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THE

PHYSICAL REVIEW

A Journal of Experimental and

THEORETICAL PHYSICS

CONDUCTED BY

THE

American Physical Society

BOARD OF EDITORS

F. BEDELL, Managing Editor

J. S. AMES E. P. LEWIS N. E. DORSEY

E. BUCKINGHAM W. C. SABINE WM. DUANE

A. A. MICHELSON A. TROWBRIDGE O. M. STEWART

Vol. X., Series II.

The Physical reviea?v

Lancaster, Pa., and Ithaca, N. Y.

1917

Pitss or

TNI NtV IRA PRINTIII* COMfAIIY
LANCASTIK. Pa.

CONTENTS OF VOL. X., SECOND SERIES

JULY. 191 7

Instability of Blectrifled liquid Snrf aces. John Zelbny z

The Magnetization of Iron, Nickel, and Cobalt by Rotation and the Nature of the Mag-
netic Molecule. S. J. Barnbtt 7

The Thermophone at a Precision Source of Sound. H. D. Arnold and I. B. Crandall . aa
A Condenser Transmitter as a Uniformly SensitiTe Instrument for the Absolute Measure-
ment of Sound Intensity. E. C. Wentb 39

The Relation of Osmotic Pressure to Temperature, n. William Francis Magib 64

Proceedings of the American Physical Society. 7a

Minutes of the Eighty-ninth Meeting; Thermal Expansion of Marble, Uoyd W,
Sckad; The Composition of Speech, /. B. Crandall; Polarization at the Cathode in
Ozsrgen, C. A. Skinner; The Energy of Emission of Photo-Electrons from Film-
coated and Non-Homogeneous Surfaces: A Theoretical Study, A. £. Hennings;
Elastic Impact of Electrons with Helium Atoms, /. Jf . Benade; Theory of Ionization
by Partially Elastic Collisions. K. T, Compton; The Passage of Low Speed Electrons
through Mercury Vapor and the Ionizing Potential of Mercury Vapor, John T.
TaU; The Kinetic Theory of Entropy; W. P. Roop; On the Ionization Potentials of
Vapors and Gases, /. C. McLennan; Optical Constants by Reflection Measurements.
L, B, Tuckerman, Jr,, and A, Q. Tool; A New Theory Concerning the Mathematical
Structure of Band Series, Raymond J, Birge; Generalized Coordinates, Relativity and
Gravitation. £. B. Wilson; The Significance of Certain New Phenomena Recently
Observed in Preliminary Experiments on the Temperature Coefficient of Contact
Potential. A. E. Hennings; Natural and Magnetic Rotation at High Temperatures.
Frederick Bales and P, P. Pkelps; A Measuring Engine for Reading Wave-Lengths
from Prismatic Spectrograms. L. G. Hoxton; The Wave-Length of Light from the
Sparlc which Exdtes Fluorescence in Nitrogen. Ckarles P, Meyer; The Necessary
Physical Assumptions Underlsring a Proof of Planclc's Radiation Law. Russell V,
Bickowsky; The Measurement of "A" by Means of C-Rays. P, C. Blake and WiUiam
Duame; The Reflection Coefficient of Monochromatic X-Rajrs from Rock Salt and
Caldte. A. H. Compton; On the Occurrence of Harmonics in the Infra-Red Absorp-
tion Spectra of Gases. W. W. Cohlentw; The Use of a Thomson Galvanometer with a
Photoelectric Cell, W, W, Coblenln; The High Frequency Absorption Bands of Some
of the ElemenU, P. C. Blake andJVilliam Duane.

AUGUST. 1917

Ionization and Excitation of Radiation by Electron Impact in Mercury Vapor and Hydro-
gen. Bbrgbn Davis and F. S. Gouchbr xox

A ReactiTe Modification of Hydrogen Produced by Alpha-Radiation. William Duanb
and Gerald L. Wbndt xx6

A Study of the Joule and l^edemann MagnetostrictiTe Effects in the Same Specimens
of NickeL S. R. Williams xa9

OKlllatory Spark Discharges between Unlike Metals. D. L. Rich X40

Optical Constants of the Binary Alloys of Silver with Copper and Platinum. Louis K.
Opprrz X56

iv CONTENTS.

Ionization of Potassiam Vapor by Ordinary Light. J. A. Gilbrsath i66

Theory of Variable Dynamical-Blectrical Syatema. H. W. Nichols 171

Proceedinga of the American Phydcal Society. 194

Radiation and Atomic Structure. R. A. MiUikan; Amplification of the Photoelectric
Current by the Audion, Jakob Kuns; High Vacuum Spectra from the Impact of
Cathode Rays. Louis Thompson; A Proposed Method for the Photometry of Lights
of Different Colors. IIL, Irwin G. Priest,
New Books 214

SEPTEMBER. 191 7

On a General Bzpanaion Theorem for the Transient Oscillations of a Connected System.

John R. Carson 2x7

The Stark Bifect in Helium and Neon. Harrry Nyquist aa6

The Ionization Potential of Blectrodes in Various Gases. F. M. Bishop 244

Internal Relations in Audion-Type Radio ReceiTers. Ralph Bown 253

Distribution of Potential in a Corona Tube. Harry T. Booth a66

The Bifect of Strain on Heterogeneous Bquilibrium. E. D. Williamson 275

Demagnetization of Iron. Arthur Whitmorb Smith 284

The Blectrical Conductiyity-of Sirattered Films. Robert W. King 291

The Mercury-Arc Pump; The Dependence of its Rate of Bzhaustion on Current. L. T.

JoNBs and H. O. Russell 301

OCTOBER. 191 7

Kinetic Theory of Rigid Molecules. Yoshio Ishida 305

Talbot's Bands and the ResolYing Power of Spectroscopes. Thomas E. Doubt 322

The Bmission of Electrons by a Metal when Bombarded by PositiTO Ions in a Vacuum.

W. L. Cheney, 335

The Fluorescence of Four Double Nitrates. H. L. Howes and D. T. Wilber 348

The Reyersal of the HaU Effect in AUoys. Alpheus W. Smith 358

Resistance and Magnetization. C. W. Heaps 366

The True Temperature Scale of Tungsten and its EmissiTO Powers at Incandescent

Temperatures. A. G. Worthing 377

Color Temperature Scales for Tungsten and Carbon. E. P. Hyde, F. £. Cady and W. £.

FORSYTHE 395

New Books. 412

NOVEMBER, 1917

Oscillating Systems Damped by Resistance Proportional to the Square of the Velocity.

J. Parker Van Zandt 41S

Theory of Crystal Structure, with Application to Twenty Crystals belonging to the Cubic

or Isometric System. Albert C. Crbhore 432

An Experimental Inyestigation of the Total Emission of X-Rays from Certain Metals.

C. S. Brainin 461

The Diffusion of Actinium Emanation and the Range of Recoil from it. L. W. McKee-

HAN 473

The Pressure Increase in the Corona. Earle H. Warner 483

The Emission of Electrons in the Selective and Normal Photo-electric Effects. A. LI.

Hughes 490

The Ionizing Potentials of Gases. A. LI. Hughes and A. A. Dixon 495

A Determination of the Planck Radiation Constant C2. C. £. Mendenhall $15

A Determination of the Ratio of the Specific Heats of Hydrogen at z8®C. and — Z90**C.

Margaret Calderwood Shields 525

Notes on Meld6's Experiment. Arthur Taber Jones and Marion Eveline Phelps . 541

CONTENTS, V

Theoretical Coneidenitions Concerning Ionization and '* Single-lined Spectra." H. J.

Van Der Bijl 546

The Parallel Jet High Vacuum Pump. William W. Crawford 557

A Determination of the Bffldency of Production of X-Rayi. Paul T. Weeks 564

The Ware Length of light from the Spark which Excites Fluoreicence in Nitrogen.

Charles F. Meyer 575

A Study of Apparent Specific Volume in Solution. Lbroy D. Weld and John C. Stein-
berg 580

The Absorption of Mercury Vapor by Tin-Cadmium AUoy. L. A. Welo 583

Proceedings of the American Physical Society. 586

Experimental Evidence for the Parson Magneton. L. 0. Grondahl.
New Books. 589

DECEMBER. 1917

Unipolar Induction and Electron Theory. George B. Pegram S9i

The Specific Resistance and Thermo-electric Power of Metallic Calcium. Charles

Lee Swisher 6ox

Total Ionization by Slow Electrons. J. B. Johnson 609

The Value of *' h " as Determined by Means of X-Rays. F. C. Blake and William

Duane 624

The Thermal Expansion of Tungsten at Incandescent Temperatures. A. G. Worthing . 638

A Single Construction for a Condensation Pump. W. C. Baker 64a

The K Series of the X-Ray Spectrum of Gallium. H. S. Uhler and C. D. Cooksey. . . 645
A Determination of the Density of Helium by Means of a Quartz Micro-Balance. T. S.

Taylor 653

A New Method of X-Ray Crystal Analysis. A. W. Hull 66z

The Critical Absorption of Some of the Chemical Elements for High Frequency X-Rays.

F. C. Blake and William Duane 697

On a Molecular Theory of Ferromagnetic Substances. Kotaro Honda and Junzo

Okubo 705

Heat Conyection in Air and Newton's Law of Cooling. W. P. White 743

Electrical and Thermal Properties of Iron Oxide. C. C. Bidwell 756

Ultrayiolet and Visible Absorption Spectra of Phenolphthalein, Phenolsuplhonphthalein

and Some Halogen DeriyatiTes. H. E. Howe and K. S. Gibson 767

Note on the Absorption of Tetrachlorophenol-phthalein. R. C. Gibbs, H. £. Howe and

E. P. T. Tyndall 779

On the Limit of Interference in the Fabry-Perot Interferometer. Megu nad Saha 78a

Errata. 787

Index. 788

Second Series. July, 1917. Vol. X., No. i

THE

PHYSICAL REVIEW.

INSTABILITY OF ELECTRIFIED LIQUID SURFACES.

By John Zeleny.

IN a recent paper^ a brief description was given of the appearance of
a liquid surface undergoing disintegration owing to instability arising
from an electric charge.

The observations recorded were made in connection with some experi-
ments on the electric discharge from liquid surfaces and the work was
confined to eye observations, through a microscope, of the surface in
question when this was illuminated by the light of a spark from a Leyden
jar. Some of the phenomena appeared to be of sufficient interest to
warrant the making of a more accurate record of them by the aid of
photography, and a few results obtained by this method are described
in this paper.

The apparatus used for getting the electrified surface was similar to
that used previously and for details reference is made to the paper men-
tioned. Briefly, the apparatus consisted of a vertical glass tube, 0.92
mm. in diameter, joined from its upper enlarged end by rubber tubing to
a reservoir of the liquid. A drop of liquid at the lower end of the small
glass tube was the part under observation. The liquid was charged to
several thousand volts from a static machine, and a grounded plate was
placed about 2 cm. below the end of the glass tube. Ethyl alcohol was
used for nearly all of the experiments inasmuch as water is not a con-
venient liquid for showing some of the phenomena, because, owing to its
high surface tension, the potential at which instability of its surface is
first obtained in air at atmospheric pressure is nearly the same as that
at which an electric discharge begins.

2. The source of light mostly employed in taking instantaneous pictures
of the liquid was a condenser discharge between magnesium electrodes in
air. For some of the exposures the spark was passed through mercury

» Proc. Camb. Philos. Soc., 18, p. 71, 1915.

JOHN ZELENY, [^S2?

vapor at atmospheric pressure in an arrangement similar to that described
by C. T. R. Wilson.i

An induction coil was used for charging the Leyden jars, and this
was provided with a rotary, mercury-jet interrupter which could be
operated successfully up to 800 interruptions per second for taking
moving pictures of the phenomena studied. In taking such pictures an
image of the drop was focused on a vertical slot in the center of a long
board placed about 2 meters from the object. A photographic film was
placed in a slide on the far side of this board, and this slide, propelled
by strong rubber bands, was shot past the opening in the board at a
speed that carried it a distance equal to the width of the opening in the
time between two interruptions. For these pictures transmitted light
and the magnesium electrodes were used exclusively.

3. Some of the photographs which were taken are reproduced in Plate
I. The magnification is not the same for all of the pictures but it can
be estimated in each case from the diameter of the glass tube which was
0.92 mm. throughout. The liquid used was alcohol except for the case
represented by Fig. 7. No luminosity accompanied any of the phe-
nomena shown so that the so-called point discharge was not present and
all transfer of electricity from the charged surface was effected by means
of droplets of liquid and none by gaseous ions.

Two series of pictures taken in the manner described in § 2 are shown
in Figs. I and 2, the time interval between the separate exposures being
approximately one eight-hundredth of a second. The pictures are to be
followed from right to left as the downward motion of the flying drops
indicates. The doubling of some of the pictures is due to irregularity in
the action of the illuminating spark. In the experiment represented by
Fig. I, the alcohol was charged to 5,000 volts and the liquid in the
reservoir was 3 cm. above the end of the glass tube, while for Fig. 2 the
corresponding numbers were 6,000 volts and 4 cms.

4. A few general statements may help to make clear what is going
on in these pictures. Suppose the liquid unelectrified and the supply
reservoir raised until liquid issues from the end of the tube at a certain
slow rate. Drops with a diameter larger than that of the tube will form,
break off and fall away. On electrifying the liquid sufficiently and lower-
ing the reservoir to keep the outflow of liquid the same as before, the
drops will be drawn out into more or less cylindrical form before they
break from the tube. After severance these cylindrical pieces will
coalesce into drops whose diameter may be considerably smaller than
that of the glass tube from which they came.

* Proc. Roy. Soc. London, 87, p. 279, 1912.

Nai^'] INSTABILITY OF ELECTRIFIED LIQUID SURFACES, 3

When however the electrification is increased to a point where the
electric force at the surface of the liquid attains a certain limiting value,
which depends upon the surface tension of the liquid and the radius of
curvature of its surface, then, irrespective of whether or not the liquid
pressure is sufficient to force liquid out of the tube, the surface becomes
unstable and any slight accidental displacement of the surface results
in a rapid increase of that displacement. This condition is first reached
at the lower end of the drop where the electric density is greatest. What
happens is that the liquid at this place is pulled out into a fine thread,
which eventually breaks up into minute drops.

5. Returning now to Figs, i and 2, it will be seen that very fine points
of liquid appear on some of the drops at the end of the tube and also
on some of the detached masses. These are the places where there is
surface instability, although the fine threads of liquid spoken of can
hardly be made out and the myriads of small droplets forming from them
are quite invisible.

The fact that instability is confined to but one very small area of the
surface is to be explained by the redistribution of electric charge caused
by the liquid drawn out from the first place that breaks down. It is
possible, however, by increasing the voltage of the surface to have a
number of these places of instability existing at the same time on a
surface of the dimensions used in these experiments.

As already explained the emission of the large drops seen in the
figures is not an accompaniment of surface instability but is conditioned
by the excessive pressure of the liquid in the tube. This pressure may
be removed by lowering the supply reservoir and then the surface of
the drop on the tube appears quite stationary with one or more of the
fine points of liquid coming quite abruptly out of the surface where
instability obtains. It is noted that the elongated detached masses of
liquid retain the instability points for a short time only after they are
separated from the tube above. The numerous droplets formed at the
points soon carry away enough electric charge to reduce the surface to
stable conditions, after which each mass of liquid quickly collapses into
a spherical drop. The rapidity with which this collapse takes place is
well illustrated by the first two pictures on the left in Fig. i. In the
second picture the elongated cylinder still carries the pointed end while
in the first picture, only one eight-hundredth of a second later, the
whole has collapsed into a nearly spherical form.

The shielding effect of the drops explains a common behavior illus-
trated by the pictures in Fig. i , where it is seen that the drops fly alter-
nately to one side and the other side of the vertical.

4 JOHN ZELENY. [to^

The electric force acting on some of the fine threads of liquid often
undergoes very rapid changes in direction on account of the drops of
liquid in front. Thus in the lower part of the sixth picture in Fig. 2
the fine thread appears as a large double loop, owing to the shielding
effect of a drop just off the picture. Smaller but more complex loops
are seen in some of the other pictures as for example in the third picture
of Fig. I. Again, the two successive pictures of Fig. 8 show a complete
reversal of field. Fig. 3 shows a case where the liquid was torn into
shreds by forces varying rapidly in direction. A number of the pieces
are barely visible, being out of focus. The appearance at the end of the
cylinder in Fig. 4 is that of clouds of finely divided material.

6. Under certain conditions of potential and hydrostatic pressure
it is possible to get the alcohol drop at the end of the tube to assume
the form of a cone with a fine thread of liquid coming from its apex.
This condition is quite steady and is especially suited for a closer study
of the liquid thread, which characterizes the state of instability, as well
as of the droplets into which the thread disintegrates. The general
appearance of this stage is shown in Fig. 5 which represents a picture
taken after an exposure of 30 seconds with light concentrated upon the
object from an arc lamp placed at the side. The potential of the drop
was 5,000 volts.

The thread or stem of liquid coming from the apex of the cone actually
had a much smaller diameter than the picture shows since it was not
perfectly stationary during the exposure. Measurements of the thread
with a microscope showed its diameter to be approximately 0.004 "^J^*
Combining this value with a measurement of the rate of emission of
the liquid it was found that the thread was pulled out at the rate of
about 8 meters per second. The liquid thread remains intact for but a
short distance in this case, breaking up into drops rather suddenly at
the place where the enlargement shows in the picture. This enlargement
is not apparent however in eye observations with a microscope, the drops
flaring out gradually from the solid stem.

The spreading of the drops formed from the central thread of liquid
into a more or less conical volume is most probably due to the combined
action of the divergence in the electric field and of the mutual repulsion
of the drops. No evidence has been obtained of any still finer threads
coming from the end of the thread visible in the picture.

The line of demarcation seen in the brushlike cloud of drops shows
these to be of two diflferent sets. The outside portion is presumably
made up of the set of comparatively small drops which form between
the main drops whenever any liquid jet breaks up into drops. None of

Na"i^*] INSTABILITY OP ELECTRIFIED LIQUID SURFACES, 5

the individual drops could be seen in flight with a microscope in instan-
taneous light, but high powers could not be used owing to their short
working distances. Rapid evaporation doubtless made futile the at-
tempts which were made to observe them after catching them on some
solid surface. The measurements given above show that even if a length
of thread equal to twice its diameter went into the making of each drop,
the number of these drops formed per second would be a million.

It may be remarked that the brush spray appears colored both by
transmitted light and by reflected light, the colors persisting in each case
when the eye is within about 45° of the direction of the light. The colors
depend not only on this angle of sight but differ also for the different
portion of the brush, indicating thus a difference between the sizes of
the drops at these parts.

It should be added that when the condition represented by Figs. I
and 2 is viewed in continuous light it too has the appearance of a more
or less compact brush, like that in Fig. 5, although the outline is not so
sharp and the appearance is somewhat granular.

7. On reducing the hydrostatic pressure below the value which was
requisite for maintaining the single central spray shown in Fig. 5, the
cone of liquid flattens into a drop and the thread of liquid now issues
from the side of this drop. Under these conditions of pressure it is
possible by increasing the voltage to obtain two points of instability
on the surface where two jets of liquid with their sprays of drops come
from the surface, as shown in Fig. 6, which is again a time exposure by
side illumination. By increasing the voltage still further eight or more
such stationary jets may be obtained at the same time, the jets being all
arranged on the outer edge of the drop.

8. Glycerine was used in some of the experiments because its viscosity
is so much greater than that of alcohol. When the conditions were
arranged for a single thread coming from a steady surface it was found
that the thread was pulled out in this case a distance of 15 mm. before
it underwent disintegration into drops. The set of large drops flared
out in their flight into a fan similar to that obtained with alcohol (Fig. 5)
but the small drops all shot out from one place in directions at right angles
to the axis of the thread and the sharp outer boundary of their paths
formed a paraboloid of revolution about this axis. The phenomenon
was not sufficiently stationary to permit a successful time exposure to be
taken by reflected light, but Fig. 7 shows a picture taken with a 2.5
seconds' exposure by transmitted light of a part of the thread coming
from the drop of glycerine. The potential used was 7,000 volts. The
diameter of this thread was approximately 0.007 ^^' ^^^ the speed with

6 JOHN ZELENY, [^SS?

which it was pulled from the drop was about 3 meters per second. The
diameter of the drops was found by catching them in various ways
and measuring them under a microscope. The large drops in the central
brush differed considerably in size but had an average diameter of about
o.oi mm., and the diameter of the small ones in the outer flare was
approximately one quarter of this value.

The long known experiment of threads being pulled from highly
electrified molten sealing wax is doubtless an example of the action
described in this paper.

I am greatly indebted to the skill of my assistant, Mr. W. B. Lang, for
the success of the pictures.

Sloanb Laboratory.
Vale Univbrsitv.

JJo^i^*] MAGNETIZATION BY ROTATION,

THE MAGNETIZATION OF IRON, NICKEL, AND COBALT

BY ROTATION AND THE NATURE OF THE

MAGNETIC MOLECULE.^

By S. J. Barnbtt.

§ I. In December, 1914, I described to the American Physical Society
an extended series of experiments completed in that year on the magnet-
ization of large steel rods by mere rotation.*

Before these experiments were made only one method of magnetizing
a body was known, viz., placing it in a magnetic field. These experiments
not only revealed another and entirely new method, but they also con-
firmed completely the fundamental assumptions on which the results
had been predicted: They proved, in a direct and conclusive way, on
the basis of classical dynamics alone, without the slightest dependence
upon the ill understood theory of radiation, (i) that Amp^reian currents,
or molecular currents of electricity in orbital revolution, exist in iron;
(2) that all or most of the electricity in orbital revolution is negative;
and (3) that it has mass, or inertia, so that each orbit behaves like a
minute gyrostat and tends to set itself with the direction of revolution
coincident with the direction of rotation impressed on the body. It is
in this way that magnetization of the body results. Furthermore, if
we admit the classical theory of radiation, these experiments, together
with the existence of residual or permanent magnetization, prove (4)
that the arrangement of the electricity in the Amp^reian orbits is Saturn-
ian rather than planetary.

§ 2. The theory of these experiments is given in the earlier paper
already referred to. If it is assumed that only one kind of electricity is
in orbital revolution, and if the mass of a particle is denoted by in and
its charge by e, it is shown that the rotation of a body with angular
velocity n revolutions per second is equivalent to putting it in a magnetic
field of intensity H, such that

* A paper read before the American Physical Society, December. 1916. A brief account
of this work is published in the Proceedings of the National Academy of Sciences, March,
1917. p. 178.

* Bamett. S. J., Phys. Rev., (2), 6, 239, 1915.

8 S. J, BARNETT. [g^

With extreme precision for all angular velocities experimentally attainable.
If electrons alone are assumed to be in orbital revolution, the second
member of this equation becomes — 7.1 X io~^ E.M.U. for electrons
in slow motion according to experiments which are well known; and
H/n should be equal to this quantity and identical for all substances.
If positive electricity also participates the magnitude of H/n should be
smaller. The mean value of H/n obtained in my 1914 experiments
was — 3.6 X io~^ E.M.U. ; and H/n was found to be independent of
the speed within the limits of the experimental error.

§ 3. Not very long after my first conclusive experiments were
presented to the American Physical Society, Einstein and de Haas,
in February and April, 1915,^ described to the German Physical Society
successful experiments on the effect converse to mine, viz., the rotation
of an iron rod by magnetization, which had been predicted and looked
for by O. W. Richardson in 1907;* and de Haas has recently continued
this work in a somewhat different manner.* Both investigations are
indirect but excellent confirmations of my own earlier work. This
work has also been confirmed by further experiments of my own of
increased precision described before the American Physical Society in
April, 1915.*

§ 4. The fundamental character of the problem, whose importance
with reference to molecular constitution is rendered greater by the
extreme difficulties encountered by the electromagnetic theory of radi-
ation in attempting to account for even the simplest cases of the
Zeeman effect and other allied magneto-optical effects, has led me to
extend the investigation, within the last year, to other specimens of
iron and to cobalt and nickel. In all the earlier work the method of
electromagnetic induction was used, a fluxmeter being the principal
measuring instrument. The new work described in this paper has
been done by the method of the magnetometer. It is more difficult
to eliminate extraneous disturbances with the new method than with
the old, but it is less difficult to attain adequate sensibility without
the use of large rods, the cost of which, in the case of cobalt and
nickel, would be great; moreover, on account of the complete novelty
of the effect under investigation, it was considered desirable to use a
method as nearly independent of the earlier one as possible.

§ 5. The magnetometer was an astatic instrument, and the rod under

^Einstein, A., and de Haas. W. J., Verh. d. D. Phys. Ges., 17, 152, 203, 420, 1915.

* Richardson. O. W., Phys. Rev., (i), 26, 248, 1908.

•de Haas, W. J., Science Abstracts A, 17, 351, 1916. The original paper has not yet
reached me.

* Barnett, S. J., loc. cit.

VOL.X.

.^]

MAGNETIZATION BY ROTATION.

experiment, or rotor, was mounted with its axis horizontal and normal
to the magnetic meridian in the equatorial position of Gauss, which
offered important advantages for this work.^ Calibrations were made by
means of solenoids wound permanently on the rotors and subsidiary
solenoids wound on wooden cores. Rotation observations were made
at equal intervals of time in sets of four as follows: The rotor was first
driven (by means of an alternating current motor) at a determined
speed in one direction, and the magnetometer scale read; then the motor
was reversed and the scale again read for the same speed ; then the read-
ings were repeated in inverse order. From the double deflection ob-
tained by subtracting the mean of the second and third readings from
the mean of the first and fourth, together with the angular velocity
of the rotor, and the calibration experiments, the quantity H/n could be
determined. The details of the experimental work and the means used
to eliminate extraneous disturbances are described below.

§6. Diagrams of important parts of the apparatus, drawn approxi-
mately to scale, are given in Fig. i, and reproductions of actual photo-

-nunec*

CMMMMTMI

N «

. vfpM fmnmmwf ^
fna

Fig. 1.

graphs are given in Figs. 2, 3 and 4. All the figures have been lettered
to correspond. In the earlier part of the work each magnet of the
astatic system was carefully made of three small pieces of tungsten steel.
In the rest of the work each magnet was made of eight steel cylinders
of very nearly the same length. All were cut from the same wire and

' Adoption of the polar position would have made it extremely difficult to make satisfactory
calibrations and to eliminate sources of serious error.

lO S, J. BARNETT. [I^S

hardened together. The two groups were magnetized in the same
field after being mounted properly on the light aluminum rod shown in
the figure. This rod carried also a small plane mirror and a thin alumi-
num damping vane, and was suspended from a torsion head A by a.
single silk fiber. The complete suspension was mounted in a groove
milled in the bronze casting Jkf, with enlargements cut for the mirror,
damping vane, and two adjustable parallel damping plates. The long
groove was covered with a strip of brass; and an opening for the mirror
and two openings for observation of the damping arrangement were
covered with glass. The enclosure was sealed with universal wax to
prevent air currents. The casting M holding the magnetometer system
was screwed to a heavy ribbed H-form bronze casting L. At its four
comers the casting L was bolted to bronze cones sunk into the tops of
the four concrete pillars 2f , 2f , K\ K\ cemented to the concrete floor.
To make the mounting more rigid, heavy boards extending from arm
to arm were screwed onto the lower surface of the H-form casting.

§ 7. Numerous experiments were made with four different rotors of
the type and dimensions indicated in Fig. 5. In constructing each rotor

*--rcm..,^ » ft

I ^1.^7 CN arlwJACM MmZM I

M ..A ir

4- 90*fCM* ••

Fig. 5.

the magnetic material was first turned to the shape indicated by the
central portion of the figure, except in the case of cobalt, where there
was a slight difference; then the bronze bearing pieces, previously
turned to the shape indicated but with centers at the ends, were soldered
to the ends of the magnetic material. Then the complete structure was
centered in the lathe and all the surfaces turned true to the same centers.
One end of a fine insulated copper wire was then soldered to one end of
the magnetic material and the wire was wound over its surface on the
lathe into a solenoid wifli 16 turns to the inch. The wire and metal
surface were then given a heavy coat of shellac and dried. Several
centimeters of the free end of the solenoid were then wound with insulat-
ing tape and covered with several layers of tin foil. The free end of
the copper wire was then stripped of its insulation and bent over this
tin foil, and a number of additional layers wound on. Then the ends
of the foil were thoroughly secured to the rotor with insulating tape.
In calibrating experiments the tin foil and the bronze bearing piece
near it were used as terminals. Before calibrating, the resistance of
the solenoid was measured to make sure that no short-circuit existed.
Two of the rotors were of cold-rolled steel shafting: One of them

Fig. 4.
S. J. BARNETT.

li^^] MAGNETIZATION BY ROTATION, II

was about 2.3 cm. in diameter and 30.6 cm. long; the other about 3.1
cm. in diameter and 30.4 cm. long. One of the rotors was nickel, about
2.2 cm. in diameter and 30.6 cm. long. Another rotor was of cobalt
about 3.2 cm. in diameter and 30.4 cm. long. The main surface of the
cobalt was a true cylinder like those of the other rotors, except that
three shallow grooves which had been turned into it were filled with thin
brass bands soldered in. Also, the cobalt casting was somewhat im-
perfect, being pitted with small holes. A fifth rotor, of Norway iron,
was constructed like the others, except that a washer was added at each
end — a fact which I discovered after finding that it failed to give satis-
factory results. Only a few rough observations were made with this
rotor.

Three of the rotors are shown at Fu Ft and F$ in Fig. 2, and one of
shown in its bearings at F in Fig. 4.

§ 8. The rotor moved in cylindrical lumen bearings, one of which,
O, is visible in Fig. 4. These lumen bearings were screwed into bronze
holders, themselves bolted into bronze castings NN, Fig. 4. The castings
NN were bolted to a single casting of bronze, which was bolted to a
heavy bronze bed plate P. The casting P was bolted to bronze cones
sunk into the oblique concrete piers S and T, Fig. 2. To assist in reducing
vibration, a considerable portion of the space between the casting P
and the piers was filled with cement and the plate bolted down before
the cement hardened. The magnetic meridian through the magnet-
ometer magnets passed nearly, but not exactly, through the center of
the rotor.

§ 9. The rotor was driven by a brass rod about 0.6 cm. in diameter
and 24 cm. long from a small bronze shaft with lumen bearings mounted
in bronze castings Q, Fig. 3, on the bed plate P. This countershaft was
itself driven by a brass rod about 0.6 cm. in diameter and 42 cm. long
from a larger bronze countershaft mounted with lumen bearings in brass
and bronze castings H, Figs. 2 and 3, on the concrete pier U. The west
end of this larger countershaft carried a three speed bronze pulley by
means of which and a similar pulley 7, Fig. 2, on the electric motor, and
a round belt, it was driven at speeds near to 20, 30 and 45 revolutions
per second. In the earliest part of the work another arrangement was
used giving speeds over 50 revolutions per second, but this was soon dis-
continued.

The electric motor was a one horse-power Century alternating current
single phase motor, and gave excellent satisfaction. On constant supply
it gave constant speeds which were identical for both directions of rota-
tion. It was reversed from a distance by simply pulling one of two

12 S, J. BARNETT. [^S^

Strings fastened to a lever which was attached to the brush holder.
Speeds were determined with a very small direct current dynamo, sepa-
rately excited at constant voltage by a storage battery, and a milli-
voltmeter connected through a high resistance and a reversing switch
with the brushes. A pulley on the armature was driven by a long belt,
/, Fig. 2, from a pulley on the main countershaft driving the rotor.
The voltmeter readings for the same speed differed slightly for the two
directions of rotation, and depended slightly on the temperature of the
field coils. In obtaining the speed from the voltmeter readings, these
effects were allowed for.

§ lo. To compensate as far as practicable for disturbances produced
by variations in the earth's magnetic intensity, a rod 5, Figs. I, 2 and 4,
called a compensator, was used. It was of the same material as the rotor,
since both were cut from the same rod in the case of each substance
investigated, and of nearly the same size, and was mounted in approxi-
mately the same position with respect to the upper magnetometer magnet
as that occupied by the rotor with respect to the lower magnet. Usually
the compensator was placed in approximately one of the symmetrical
positions 5, 5', Fig. I, but the best position had to be found by trial.

§ II. Although the earlier investigation on iron by the method of
electromagnetic induction had shown that the rotation of the rotor in
the earth's magnetic field gave the same or nearly the same results as
were obtained when the intensity of this field was annulled by a suitable
electric coil, it was considered important for the present investigation to
provide means of neutralizing the earth's field throughout the region
occupied by the rotor. For this purpose the large and accurately made
coil, of rectangular cross-section, used in the last part of the earlier
investigation was slightly modified. The frame work was shortened
along the axis of the coil, and strengthened and made still more nearly
true by brass bolts and internal wooden braces near the central section.
The coil was reduced to 56 cm. in length, and was left about 26 cm. wide
and 198 cm. broad. The coil and frame are marked with the letter E
in Figs. 1-4.

As in the earlier experiments, the coil was mounted over the rotor
and its oblique piers 5 and T, the centers of coil and rotor being made
nearly coincident. The position of the frame was adjusted until the
long edges were horizontal and perpendicular to the magnetic meridian,
and the axis of the coil was parallel to the earth's intensity. Then the
frame was bolted to bronze cones sunk into six concrete piers, three on
each of the larger sides. The three piers on the north side are shown at
RRRm Fig. 2.

JJS"x^] MAGNETIZATION BY ROTATION, 1 3

After the coil had been clamped in position, with the heavier bronze
castings in place, its magnetic field, when it was traversed by an electric
current, was studied throughout a region including and extending some-
what beyond that to be occupied by the magnetic part of the rotors.
This field was found to be uniform to one part in five hundred. By a
method similar to that used in the earlier experiments, it was found
that a current in the proper direction giving 914 divisions (equivalent
to about 0.389 ampere) on a Weston instrument (with special shunt)
compensated completely the earth's flux through a steel rod 33 cm.
long and 3.2 cm. in diameter with center in the position to be occupied
by the center of each rotor. The compensation was sensitive to a tenth
division, or about one part in nine hundred. After the rotation experi-
ments were completed the compensation was again tested by the same
method and was found to have remained unaltered. The current
was kept at the compensating value 91.4 divisions during nearly all
of the observations, and was always kept within one tenth, or in a few
instances two tenths, of a division of that value, except for testing
purposes as indicated below.

The concrete piers and all the castings near the rotor were free from
iron, and all the other bronze and brass castings were either free from
iron or so nearly free that any effect on the field in the region occupied
by the rotor was quite negligible.

§ 12. Since the lower magnetometer magnet hung, as seen from Fig. i,
in a region in which the earth's intensity was nearly annulled, it was
necessary, in order to keep the magnetometer sensibility and zero reading
approximately independent of the current in the compensating coil,
to provide special coils CC^ Figs. 1-4, to compensate approximately the
horizontal component of the earth's intensity in the region occupied by
the upper magnet. Each of these coils contained three turns of insulated
wire. They were wound in vertical planes on brass rings 10 cm. in
diameter, whose bases were moved for adjustment in brass slides by a
right and left handed screw in such a way that the magnet was always
approximately at the center of the system. The coils were connected
in series with the main compensating coil, and both were connected to
oppose the earth's intensity. The distance between the rings was ad-
justed in the different experiments until the earth's horizontal intensity
at the upper magnet was nearly annulled.

The control magnet Z>, Figs. 1-3, was a small piece of hardened tool
steel.

With the arrangement described the sensibility of the magnetometer
was not altered when the compensating current was changed by one part

14 S. J, BARNETT. [ISS

in ninety, from 91.0 to 92.0 divisions on the indicating instrument,
and the zero was altered but little.

§ 13. The magnetometer was almost always used with approximately
critical damping, when it usually required from 15 to 20 seconds to reach
its elongation.

The mirror was a small plane mirror and the opening in front of it
in the bronze holder was covered with a convex spectacle lens. Deflec-
tions were read to tenths of millimeters by means of a single filament
tungsten nitrogen-filled lamp and a translucent scale, distant about 4 m.
from the mirror except in the very earliest part of the work.

§ 14. Calibration experiments were of two kinds. At the beginning
and the end of a series of observations, except in rare instances, the
approximate sensibility was carefully determined by the process described
below; and on a single occasion for each rotor the correction necessitated
by the presence and finite length of the solenoid permanently wound
upon it was determined once for all. This correction was much less
than the experimental error, but was nevertheless always made.

The approximate calibration for each series of rotation experiments
was made as follows: A dry cell in good condition, with open circuit
E.M.F. 1.50 volt, was connected through a suitable key and a standard
high resistance — 7,500 to 25,000 ohms — in series with the solenoid of
the rotor under experiment, and magnetometer deflections on opening
and closing, or (usually) double deflections on reversing, the key were
obtained. Each solenoid, as stated above, was wound by lathe with
16 turns to the inch. If D denotes the double deflection, R the box
resistance in the circuit (that of battery and solenoid being negligible) in
ohms, and h the magnetic intensity which the solenoid, if very long,
would impress on the rotor, and if d denotes the double deflection
produced on reversing the rotor in the rotation experiments, the intrinsic
magnetic intensity of rotation is approximately

__ d . 47r X 16 X 1,50 d

H = T:h = — — ^^ p r^ gauss;

D 10 X 2.54 X R D^

and, if n denotes the rotor velocity in revolutions per second, the intrinsic
intensity per unit velocity is approximately

H ^ 47r X 16 X 1.5 X d gauss

n 10 X 2.54 X R X D X n rev. per second *

Since the magnetometer zero and sensibility depended on the position
angle of the rotor, the mean value of D for three position angles differing
successively by 120° was obtained in all the later experiments. The
same value of the mean was obtained for three position angles half way
between those just mentioned. In the case of the earlier observations

)J^,f] MAGNETIZATION BY ROTATION, 1 5

on the larger rotor of steel and the rotor of cobalt, the omission of this
precaution introduced a possible error of 2 or 3 per cent., which, however,
is much less than the experimental error in the rotation experiments.
In the case of the nickel rotor the error introduced in this way was only
a half of one per cent., or less; and the same thing would be true of the
smaller steel rotor, which, however, was always calibrated with the
three or six position angles.

The calibrations at the beginning and end of a series always agreed
closely. All calibrations were made with the proper current in the com-
pensating coils, although, as already stated, considerable variations of
the compensating current did not effect the sensibility appreciably.

§ 15. The experiment to correct for the departure from uniformity
of the field produced by the rotor's solenoid and for the effect of the
solenoid itself were most conveniently made with the rotors and coils
near the upper magnet instead of the lower magnet.

In the case of each rotor it was found that adding to each end a solenoid
10 cm. in length, wound like the rotor but on a wooden core of approxi-
mately the same diameter, made no difference in the deflection produced
by a given current.

The total length of the combined solenoids was about 50.5 cm. Sole-
noids of approximately the same diameters as those of the rotors and
wound jn the same way, but on wooden cores 50.5 cm. in length were
mounted symmetrically in the place occupied previously by the rotors,
and the deflections produced by known currents observed. From the
ratio of the currents, and the ratio of the magnetometer sensibilities
with rotor present and rotor absent, which precautions were taken to
obtain, the ratio of the deflection produced by the solenoid alone to
that produced with the same current by the solenoid and the rotor
together was obtained for each rotor. The corrections thus found were
1.2 per cent, for cobalt; 0.6 per cent, for nickel; 0.9 per cent, for the
smaller steel rod; and 1.3 per cent, for the larger steel rod.

Experiments were also made as a check and as a matter of interest
with the central 30.5 cm. of the solenoids wound on wooden cores in place
of the full lengths. Corrections obtained in this way are almost exactly
twice those with the larger solenoids, which, being the true values, were
applied to the observations.

§ 16. If C denotes the per cent, correction obtained above, and Z>o the
corrected calibration double deflection, the true value of Hjit will be
obtained by substituting for D in the final equation of § 14 the quantity

Z>o = Z>(i -— ).
\ 100/

5. J, BARNETT.

[Sbcond
r

Thus we get, to a close approximation,

^ 4T X i6 X 1.50 Xdx(i+~-)

n "" 10 X 2.54 X RX n X D rev. per second *

gauss

(2)

§ 17. The chief results of the observations are given in Table I. The
sets (§ 5) are arranged in groups, each group containing from 2 to 14
sets, all at very nearly the same speed. The last two columns contain,
for the series of observations occupying each horizontal row, the average
departure of a single set from the mean value given in column 6 reckoned
in two different ways. The value in column 6 is the weighted arithmetic
mean calculated by assigning to the mean for each group a weight
proportional to the number of sets in the group.

To obtain the departure given in the next to the last column the pro-
cedure was as follows: For each group of the series the average departure
from the group mean was determined. This was multiplied by the
number of sets in the group, and the sum of the products so obtained
for all the groups in the series was divided by the total number of sets
in the series. The departure given in the last column was obtained by
taking the difference between the mean value given in column 6 and each
of the group means, multiplying each difference by the number of sets
in the group, adding and dividing by the total number of sets. The
two columns together give a sufficiently good idea of the experimental
errors.

§ 18. In addition to the observations given in the table, a few observa-
tions were made with the larger steel rod at lower speeds when conditions

Table I.

Intrinsic magnetic intensity of rotation in iron, nickel and cobalt.

Rotor.

Series.

Qroupt.

Mean
Speed
R.P.S.

Steel (smaller) .
Steel (larger) . .
Cobalt

1
2
3

1-2

3-4
5-7

44.8
47.8
20.2

8-11

30.3

12-25

45.5

45.0

44.8

Nickel

20.5

27-28 ,

30.5

29-32 1

45.3

Number
of Sett.

B.M.U.
Mean.

5.1

5.2

4.8

5.6

6.0

6.5

5.9

6.1

4.7

6.7

6.1

Averai^e Averagre
Departure ' Departure
from Mean, from Mean

(Seta). I (Qroupa).

0.5
1.2
2.2
1.2
0.9
0.3
0.4
0.4
2.0
1.1
0.5

0.5
0.6
2.2
1.4
0.8

1.1
0.9

X2J"i^'] MAGNETIZATION BY ROTATION. 1 7

were such that the extraneous disturbances largely masked the effect
being looked for. Moreover, a few sets were taken with a fifth rotor, of
soft Norway iron, which gave results consistent with those given in
the table, but with large discrepancies in the magnitude of the deflec-
tions. This rotor was very troublesome, and was found, after the experi-
ments were completed, to have been constructed differently from the
others — accurate washers having been added at the ends because the
rotor had been found too short to fit the bearings perfectly. This
construction may explain a part, but will not probably explain all, of
the rotor's behavior, which has not yet been adequately investigated.
Furthermore, on one occasion several sets with the nickel rod gave dis-
cordant deflections several times as great as the normal deflections for
the same speed. On examination it was found that the rotor had been
improperly mounted with large longitudinal play, suggesting such an
effect as is obtained by tapping an iron rod while in a magnetic field.
On another occasion, just after the completion of a long and good series
of observations on nickel, with normal compensating current, at the
end of a night's work, three sets of readings were taken with the com-
pensating current above and below normal value. All were discordant
and abnormally low. These and the other observations mentioned
were omitted from the table.

§ 19. Six of the observations in series 11 on nickel were made with
compensating current above and below the normal value, at 91.0 and
92.0, instead of 91.4, divisions. They are all included in the table,
however, because the alterations of the current produced no change.
With the rods of iron and cobalt the change produced by altering the
current as in the case of nickel, from 91.4 to 91.0 and 92.0 divisions,
was but little if any greater than the experimental error; but the observa-
tions are not included in the table because of the difference. In the
case of cobalt the observations were taken on the same occasion with
those for group 25. Group 25, with current 91.4 divisions, gave the
deflection i.oo±o.o6 cm. for 5 sets. The same number of sets for
currents 91.0 and 92.0 gave deflections 1.13 ± 0.04 cm. and 0.96 ± 0.06
cm., respectively.

§20. The observations mentioned in the last section show that no
appreciable systematic error was introduced on account of currents
induced in the rotor by its motion in the field of the earth, compensated
as it was to about one part in nine hundred.

The field intensity produced at the center of the rotor by the control
magnet was about one one thousandth the earth's intensity. The intensity
produced by the lower magnetometer magnet at the same point was

1 8 S. J, BARNETT. [toS

about equal, and had always a large component opposite, to that due to
the control magnet. The intensity at the rotor due to the upper
compensating coils was about one three thousandth the earth's intensity;
and that due to the magnetization, both permanent and temporary, of
the compensating rods, was also negligible.^ Hence it would be un-
reasonable to suppose that any appreciable systematic error was pro-
duced by the motion of the rotor in these fields. That no great error
of this sort was introduced is proved experimentally, moreover, by the
agreement of the results obtained with the two rotors of iron of different
diameters, inasmuch as any eddy current effect would depend upon the
diameter.

§ 21. Another possible systematic error which had to be avoided was
the error arising from the shift of the rotor's axis in azimuth or altitude,
the shift being probably different for the two directions of rotation.
If the residual field intensity normal to the axis of the rotor is Z and
the maximum angular shift possible on reversal a, the maximum change
of longitudinal intensity impressed on the rotor by an angular displace-
ment is Za, The difference between the internal diameter of the lumen
journals and the diameter of the bronze bearings was about 0.004 cm.,
and the distance between the far ends of the journals was about 35 cm.
For the maximum possible value of a these data give (2 X 4)/35,ooo.
If we assume that Z is as great as 1/500 the earth's intensity, or about
0.6/500 gauss, we obtain as the maximum value of aZ the quantity
(8 X 6)/(35,ooo X 5,000) gauss, or about 3 X lO"' gauss. This inten-
sity, which is certainly greater than any intensity of the sort which could
have been produced, is only about one fourth the change of intensity
which would be produced in the rotation experiments, by reversing the
direction of rotation at a speed of one revolution per second. Any such

^ Before beginning experiments with the smaller iron rotor, and before making the later
experiments with the other rotors, their compensating rods were heated to whiteness and
otherwise treated to demagnetize them as thoroughly as practicable. On making tests with
a magnetometer, after the rotations were concluded, it was found that the maximum mag-
netic intensity which the temporary diametral magnetization of any of the compensators,
placed in the undisturbed field of the earth, produced at a distance somewhat less than the
normal distance between compensator and rotor was about one sixteenth hundredth of the
earth's intensity. The actual intensity produced in the region occupied by the rotor during
the rotation experiments was much less than this, since the compensator was then in a field
of reduced intensity and since the plane containing the axes of rotor and compensator made
a considerable angle with the earth's intensity. The maximum intensity in the region
occupied by the rotor due to the permanent diametral magnetization of any compensator
was found to be less than one ten thousandth of the earth's intensity; and the maximum
intensity due to the permanent longitudinal magnetization was found to be about the same.
The agreement between the results obtained before and after the compensators received the
treatment described shows that the effects of the compensators were negligible in the early
part of the work as well as in the later.

}J^,f] MAGNETIZATION BY ROTATION. 1 9

effect in these experiments was therefore negligible. So far as angular
displacements of the rotor's axis in the plane parallel to the largest side
of the compensating coil are concerned, this is also proved experimentally
by the observations mentioned in § 19.

§ 22. Possible systematic errors due to the longitudinal motion of the
rotor, carrying its magnetization with it and undergoing changes of
magnetization on account of the space variation of the longitudinal
components of the residual field intensity, were avoided by mounting
the rotor free from appreciable longitudinal play and observing always
only the effect of reversing the angular velocity. For a given speed
there is no reason to expect a different longitudinal displacement, if any
should occur, on reversal of the direction of rotation. Error due to the
bodily motion of the magnetization with the rotor would also be elimi-
nated in part by the process mentioned in the next section.

§ 23. As follows from the earlier investigation on iron no error due to
torsion was to be expected. Nevertheless, the rotors were made rever-
sible in their bearings, and in the cases of nickel and cobalt many sets
of observations were made with the rotor turned in each direction, a
process which would eliminate the torsion error if existent. No difference
was found.

§ 24. Although the rotors were demagnetized until the residual longi-
tudinal magnetization was in no case greater than about one tenth that
of the principal rod of iron used in the earlier investigation, it always
happened that when a rotor was rotated very slowly by hand the image
of the lamp filament moved up and down on the scale — over many
centimeters in the case of the cobalt and Norway iron rotors, and over
smaller ranges in the case of the others. This is one of the reasons for
the necessity of always making observations while the rotor was in motion.
The procedure adopted of obtaining the difference of scale readings for
both directions of rotation at the same speed avoided difficulty from
this source. This procedure also avoided error due to the change of
magnetization by centrifugal expansion of the rotor, discovered in iron
in the course of the earlier investigation. This effect was doubtless
much smaller in these experiments than in the earlier ones, as the residual
magnetization was much less, but the method of observation did not
permit its examination.

§ 25. All bearing parts were carefully turned and adjusted, and were
oiled almost invariably before each set of four readings. These pre-
cautions, with the heavy mountings and special method of driving al-
ready described, eliminated almost completely, if not completely, mechan-
ical disturbances due to the rotation. Other mechanical disturbances

20 S. y. BARNETT. [ISSSu

and magnetic disturbances were reduced as much as possible by beginning
work in nearly all cases after one o'clock, and quitting before four o'clock,
or sometimes a little later, in the night. Except during a few sets on
one night, when a high wind was blowing, mechanical disturbances
were never troublesome; but magnetic disturbances, in spite of the pre-
cautions taken to secure an astatic system and to adjust the magnetic
compensator, were always present; and they account for the chief
part of the accidental experimental error. The temperature during
the night work was usually very nearly constant, and the compensating
current varied very slowly, often requiring no adjustment for many
sets. The speed of the driving motor also remained very nearly constant.

§ 26. In order to avoid all extraneous disturbances as far as possible
the method of observation already described (§ 5) was adopted and
was carried out on a regular time schedule. All being in readiness, the
motor was started at a certain time T. After a fixed interval of t seconds
(usually either 15" or 20") the magnetometer scale and speed voltmeter
were read, and the motor then stopped and the motor and voltmeter
switches thrown for reversal. At the time Z" + i "* the motor was
started in the opposite direction, and the readings taken t seconds later
as before. Then observations for the two directions of rotation were
made in inverse order, the motor being started at the times T + 2°^
and T + 3"*, and the readings being taken in each case t seconds later.
The magnetometer double deflection obtained by subtracting the mean
of the second and third scale readings from the mean of the first and
fourth was independent of any slow drift and corresponded to the mean
of the four speeds, always close together. In a few sets the constant
interval between successive observations differed from i"; in a few
the interval between the second and third differed from the other inter-
vals, which was legitimate; and in some cases sudden magnetic disturb-
ances made it necessary to observe the scale at a time differing from the
schedule time; but the usual procedure was that given above, and
departures from it were unimportant.

§ 27. With nickel and cobalt observations were made at three speeds.
As shown in Table I., H/n was found to be independent of the speed
within the limits of the experimental error, a result already obtained in
the earlier experiments with iron. Since the chief disturbances were
magnetic, the observations at lower speeds were less precise than those
at the highest speeds. The results at the highest speeds are given in
series i, 2, 5 and 11. Series 6, 7 and 8 are a part of series 5, viz., the
last three groups of results obtained with cobalt in a neutral field, one
group of 5 sets (group 23), obtained shortly before group 24 while a

X^jf] MAGNETIZATION BY ROTATION. 21

Strong wind was blowing and the magnetometer was imperfectly damped,
being excepted.

§ 28. Every set of observations^ gave the sign of H/n negative like
that of ^Trmje for an electron. The mean magnitude of HIn is in all
cases somewhat less than the accepted magnitude of ^Trm/e, viz.,
7.1 X lO"' E.M.U., obtained from other experiments on electrons in
slow motion, ranging from 5.1 to 6.5 X io~^ E.M.U. for the most reliable
observations in Table I., viz. those at the highest speeds. The differ-
ences are in the same direction as in the earlier experiments on iron, which
gave 3.6 and 3.1 in place of 7.1 ; but the experimental errors, on account
of the great difficulties involved, are such that importance cannot in
my opinion be attached to the discrepancies. The investigation must
rather be taken as confirming equation (i) both qualitatively and
quantitatively on the assumption that only electrons are in orbital revolu-
tion in the molecules of all the substances investigated. It shows more-
over that the effect is independent of the size of the body in rotation,
which is an implicit requirement of equation (i).

§ 29. This investigation has been made with the aid of a grant from
the university for which I am indebted to the interest of the dean of the
graduate school, Professor Wm. McPherson. I am indebted to Mr.
Arthur Freund, mechanician in this laboratory, for most of the finer
mechanical work necessary; and I am indebted to Mrs. Barnett for a
great deal of help in making the experiments.

Thb Physical Laboratory,
Ohio State University,
March 13, 1917.

» Except one, at a low speed, among the early observations mentioned in § 18. in which
the diiscrepancies were great and the effect was reached by extraneous disturbances.

22 H. b, ARNOLD AND I, B, CRANDALL, fSSSI?

THE THERMOPHONE AS A PRECISION SOURCE OF SOUND.

By H. D. Arnold and I. B. Crandall.

nr^HE acoustic effect accompanying the passage of an alternating
•*• current through a thin conductor has been known for some time,
but as far as we are aware, no use has been made of the principle involved
for the production of a precision source of sound energy, or standard
phone. In 1898 F. Braun^ discovered that acoustic effects could be
produced by passing alternating currents through a bolometer in which
the usual direct current was also maintained. An artick by Weinberg*
describes the old experiments of Braun, and also some experiments of
Weinberg, in which acoustic phenomena were observed with other
'electrically heated conductors, rheostats, etc., through which large alter-
nating currents were passed. A more recent application of the same
principle is described by de Lange' in his article on the thermophone.

The writers have found that the thermophone together with a suitable
'supply of alternating current can be used very conveniently as a precision
source of sound energy. On account of the fact that the published
material on this electrical-acoustic effect is largely of a qualitative
character it has been necessary to work out a quantitative theory;
and it is the purpose of this paper to give the theory and show how the
instrument can be adapted to acoustic measurements.

When alternating current is passed through a thin conductor, periodic
heating takes place in the conductor following the variations in current
strength. This periodic heating sets up temperature waves which are
propagated into the surrounding medium; the amplitude of the tempera-
ture waves falling off very rapidly as the distance from the conductor
increases. On account of the rapid attenuation of these temperature
waves, their net effect is to produce a periodic rise in temperature in a
limited portion of the medium near the conductor, and the thermal
expansion and contraction of this layer of the medium determines the
amplitude of the resulting sound waves. To secure appreciable ampli-
tudes with currents of ordinary magnitude it is essential that the con-

» Ann. der Physik. 65, 1898, p. 358.

* Elektrot. Zeit. 28, 1907, p. 944. See also A. Kocpsel, Elektrot. Zeit. 28, 1907, p. 1095.

* Proc. Royal Soc. 91 A, 191 5, p. 239.

VOL.X

«^]

THERMOPHONE AS A PRECISION SOURCE OF SOUND.

CLAMP

CURRCNT
LEMD

nom

PumNUM Strip

.00007 CH.TWCK

ductor be very thin; its heat capacity must be small, and it must be
able to conduct at once to its surface the heat produced in its interior,
in order to follow the temperature
changes produced by a rapidly varying
current.

A simple form of instrument which we
have used is shown in Fig. i. There
are two ways in which the strip may
be supplied with electrical energy in
order to produce sound waves, (a) with
pure alternating current and (i) with
alternating and direct current superimposed. If an alternating current
/ sin pt is supplied, the heating effect is proportional to

Fig. 1.

Simple Thennophone.

RP sin2 pt ^ — (i - cos 2pt),

(I)

so that the acoustic frequency is double the frequency of the applied
alternating current. If it is desired to make the acoustic wave follow
the alternating current wave, without introducing the double frequency
effect, resort must be had to a superimposed direct current whose value
is several times as large as the maximum value of the alternating current.
If a direct current /© and an alternating current /' sin pt are used to
heat the strip, the heating effect is proportional to

i?(/o + /' sin pty = Rh^ + 2Rhr sin pt + RF^ sin^ pt

= i? (/o« + y) + 2Rhr sin pt -

cos 2pt

(2)

from which it is evident that the double frequency term can be made
negligible by suitable choice of /o and /'.

When pure alternating current is used, the mean temperature of the
strip is determined by the term \RP\ when direct current is used with
the alternating current, the mean temperature is determined by the term
RI^. The mean temperature of the conductor is one of the factors
which sets a limit on the maximum amount of electrical energy used and
hence on the maximum amount of acoustic energy that can be obtained.
If only a small quantity of alternating current energy of suitable fre-
quency is available, it is clear, from a comparison of equations (2) and
(i) that more acoustic effect will be realized if direct current energy is
added up to the limit that the strip will bear; for example, if /o* is as
large as /'^, the product term in (2) is four times as large as the second
term in (i).

24 H. D. ARNOLD AND L B. GRAND ALL, [ISSS

Suppose now that an indefinite quantity of alternating current energy
of any frequency is at hand ; we desire to find the most effective way to
actuate the element. Equating the terms in (i) and (2) which are
proportional to the (limiting) mean temperature in each case

[fL-[M^.-?)L

nuu ■• ' ' -"max

we can compute the maximum amplitude 2RI0I' of the product term in
(2) and compare this with the amplitude RP/2 of the periodic term in
(i). The maximum value of the product 2RI(ir, consistent with condi-
tion (3) is R1}1^2 and implies the relations

/= v/Jr = 2/0.

The amplitude RPI>^2 in the second case is only slightly larger than the
amplitude RPI2 which we should have according to (i); and in the
second case there is the double frequency term of amplitude RPI^
which in most cases would be inconveniently large.

The conclusion from these calculations is that for sounding a pure
tone of a given frequency it is better to actuate the strip wholly by
alternating current of half that frequency. However, if it is desired
to make the sound waves reproduce the electrical waves in both frequency
and form, it is necessary to use in addition a direct current whose relative
value is large. In this case the thermophone element is worked somewhat
below maximum efficiency for the sake of minimizing the double-fre-
quency effect.

Using the first method of excitation, it is necessary, if a pure tone is
desired, that the alternating current used be a pure sine wave, absolutely
free from harmonics. In order to show the acoustic effect of harmonics
in the alternating current supply, consider an exciting current of the
form

2^ a* sin kpL
The heating effect produced is proportional to
]C o.k sin kpt) =22 a J? sin^ fe/>/ + 52 52 a^* sin jpt sin kpii

= I2-T — 22— cos 2kpt + 52 22-T^cos o* - *)/>*

- 52 52-^- cos 0' + k)pt

>=1 Jbs2 2

Na*if*] THERMOPHONE AS A PRECISION SOURCE OF SOUND. 25

which shows that two series of combination- tones result in addition to
the series of tones whose frequencies are double those of the applied funda-
mental and harmonics. One particular case is of practical importance:
the case in which the alternating current wave consists of a fundamental
and an appreciable second harmonic. In this case, besides the tones of
double and quadruple frequency there are combination tones of single
and triple frequency, a paradoxical result that is very easily verified by
experiment. The importance of a pure alternating current supply is
clear from the considerations given.

The Periodic Temperature Change in a Thin Flat Conductor

Supplied with Alternating Current.

Consider first the case of a strip supplied with both direct and alter-
nating current. Equating the rate of production of heat by the electric
current to the rate of transfer of heat to the surrounding medium, plus
the rate of storage of heat in the strip, the fundamental equation may
be written:

dT
o.24(/o + r sin ptyR = 2apT + ^7 -^ . (4)

in which the unit is the calorie per second, and the constants are chosen

as follows:

/o = direct current in amperes.

/' = maximum value of A.C. in amperes.

p = 2t/; / = frequency.

R = instantaneous resistance of the strip.

T = temperature of strip above surroundings.

a = area of one side of strip.

/3 = the rate of loss of heat per unit area of the strip (due to conduc-
tion and radiation) per unit rise in temperature of the strip
above that of its surroundings; it is equal to the product of the
temperature gradient per degree rise, into the conductivity of
the medium. It can be determined experimentally, and is a
constant if only conduction is considered; if it is desired to
take account of radiation a modified value of /3 for any value of
T may be obtained which is sufficiently accurate for the purposes
of calculation. The rate of radiation is not great at low tempera-
tures, and only becomes equal to the rate of conduction at about
500^ C.

ay = the heat capacity of the strip, 7 being equal to the product of the
thickness of the strip by the specific heat per unit volume.

26 H. D. ARNOLD AND /. B. CRANDALL. [toSS

The factor is analogous to the mass in vibratory mechanics, and
the inductance in alternating current calculations.
The equation for To, the mean temperature above surroundings is:

0.24 ( hm + — j = 2afiTo. (4^)

Combining equations (4) and (4a) we have the following, which
contains only factors which vary with the time:

/ /'* V dT

0.2422 ^2/o/' sin pt cos 2pt\ = 2ap{T - To) + ^7 77 • (S)

In obtaining a solution for T — To we shall neglect transient effects,
also the double frequency term. The double frequency effect is the
principal effect in the case of a pure alternating current supply as given
below ; but here we simply remark that we can make the double frequency
term as small as we please by a suitable choice of the ratio /o : /'.

The solution of the equation

dT
,4SRIoI' sin pt = 2ap{T - To) + ^yj7 C^)

is, neglecting transient effects,

i — i 0 = — 7--^^ - --=- sm I pt — tan * — ^ I , (7)

which gives the periodic temperature variation of the strip. Note that
if i is the effective (measured) value of the A.C., v''2 i must be written
in place of /' in (7).

If the strip is supplied with alternating current only, the fundamental

equation becomes

dT
o,24RP sin2 pt = ,i2RP{i - cos 2pt) = 2apT + ^y-JT - (4 )

The mean temperature in this case is defined by

.12RP = 2apTo (4'a)

and the differential equation which T — To must satisfy is

dT
.12RP cos 2pt = 2ap{T - To) + ay — . (5O

The solution of this equation is, neglecting transient effects

,12RP

T - To-=

-cos(2/>/-tan-»^). (7')

2a^^^ + y^i^ ^ 2)8

Having found the magnitude of the temperature variation in the
strip, we go on to calculate the magnitude of the effect in the surrounding
medium.

no*!^'] thermophone as a precision source of sound, 2 7

Theory of the Effect in the Medium.

Consider an infinite plane metal plate with a column of gas extending
normally from either face of a certain portion of the plate; this is equiva-
lent, mechanically, to the strip conductor if terminal conditions are
neglected. If the temperature of the plate is a sine function of the time,
temperature waves will be propagated into the atmosphere on either
side; and calculation will show that these waves are so heavily damped-
that they are practically extinguished after one wave-length has been
traversed. Within this region there is a rise and fall of temperature of
the medium with every cycle, and the resulting expansion and contrac-
tion of this narrow film of the medium near the source accounts for the
sound vibration produced.

In the derivation of equations (7) and (7O it has been tacitly assumed
that no electrical energy was spent in expanding the strip, as this effect
would be relatively very small. It is evident from conditions of sym-
metry that there is no force on the strip tending to make it vibrate;
hence no energy can be used mechanically. In calculating the effect
on the medium we shall consider two cases:

1. In which the periodic rise in temperature of the strip is allowed
to produce a continuous stream of sound energy, propagated away from
the strip as plane waves. It is an easy matter to modify this treatment
to fit the case of diverging waves in the open atmosphere.

2. In which the strip is placed in a small cavity for the purpose of
producing pressure changes; these pressure changes being used to
actuate the ear, or $ome mechanical phonometer which constitutes one
wall of the enclosure.

The reason for giving separate treatment to these two types of action,
is that in the first case we can speak of a definite amplitude and particle
velocity, and a corresponding propagation of energy; whereas in the
second case, amplitude and velocity are indefinite terms, and pressure
change is much more readily calculated. It is by virtue of pressure
change that the acoustic energy generated makes its effect on the bound-
ing wall,. and if the dimensions of the cavity are small compared to the
acoustic wave-length, the pressure change produced at the strip is
quickly distributed over the whole enclosure.

First Case: Wave Propagation from the Strip. — ^Assume that the
periodic temperature variation in the strip results in the expansion and
contraction against constant atmospheric pressure of a certain layer of
air next to the source. This implies that the very small pressure changes
that do arise at the boundary of the layer (as the result of rapid change
in volume) are propagated into the atmosphere with such high velocity

H, D, ARNOLD AND /. B, CRANDALL.

rSBcoifi>
Ir

that they do not react appreciably on the expansion of the layer. This
condition is realized in practice because the velocity of sound in air is so
much greater than the velocity of the vibrating boundary which produces
the sound.

In treatises on the conduction of heat it is shown that the temperature
at any point of the medium distant ± x from a plane source of tem-
. perature, varying periodically as in equation (7), may be expressed as the
following function of space and time:

r.' = re-^ sin (J>t±ax), (8)

in which a = ^p/ik, p being 2tX frequency, and k the **diffusivity"of
the medium, or the ratio of the thermal conductivity to the specific
heat per unit volume. The value of this constant for air at o® centigrade,
using the specific heat at constant pressure is 0.17 C.G.S. units.

It is necessary to know the effect of the temperature of the medium
on k and this can be found by considering separately the conductivity
and the specific heat. The former is proportional to the square root of
the absolute temperature; the specific heat per unit mass is practically
independent of temperature thus making the specific heat per unit
volume proportional to the reciprocal of the absolute temperature.
Since k is the ratio, we may write

* = o.i7(-) (9)

nsuMOti

T^.T'e^wCiJt^ where 6 denotes the absolute tempera-

ture of the medium.

The velocity of propagation of the
temperature wave is, from (8)

vvy

= ^ = \/2pk
a

and the wave-length

Cumc

HOMYCMRtK

mnvoNxAx

A
B
C
D

-0

-aoaor

OiO

♦oooor
♦aiiiT'

Sir^k

(10)

Fig. 2.

If the wave-length is taken as a
unit,^it is easy to plot the course of
Tm as a function of x for any given time, t, as is shown in Fig. 2. This
shows clearly the enormous damping of^these waves of acoustic frequency;
it also shows that practically all of the expansion effect due to periodic
rise in temperature takes place within^the region bounded by the plane

a: = X = — .
a

Na"xf'] THERMOPHONE AS A PRECISION SOURCE OF SOUND. 29

In order to compute the amount of the periodic expansion, we desire to
know the mean value of the temperature rise in this region as a function
of the time: that is,

— I c~** sin {pt — ax)dx

t. e., its maximum value is .ii2T'^ and it lags the varying temperature

of the strip by the angle T/4.

If the mean absolute temperature of the air film is B, the maximum

expansion will be,

dB V

-T = .1 12 — per unit volume, (12)

or per unit length, if expansion is considered to take place in only one
direction. The length in question is a wave-length, and this by equa-
tions (9) and (10) is, at B^

X = .xJ| = o.8aJ^'(^f. (X3)

Multiplying (12) and (13) we obtain for the absolute increase in length
due to expansion

- 16 r

This may be considered as the maximum ** amplitude" of a sound wave
leaving the plane a: = X, if the effect of the expanding and contracting
air film on the surrounding air is the same as that of a solid moving
piston — assuming also that the amplitude of the sound produced by a
moving piston is equal to the amplitude of the motion of the piston
itself.

If the thermal conductivity were proportional to the first power of the
absolute temperature, instead of to the square root, we should have had,
instead of (12)

o.i6r

^/ .273

The departure of (14a) from (14) is not serious, if the temperature of
the film is 300° C. or less, as in this case ^^^*«(273)''* is less than 330.
The air film is always considerably cooler than the strip, so that the
strip might have a temperature of (say) 500® without causing more than
a 20 per cent, discrepancy between { and {1.

30 H. D, ARNOLD AND I, B. CRANDALL, [&S»

In order to have the amplitude of the sound wave in terms of the alter-
nating current supplied to the strip, we make use of equations (7) and
(7') which give the variation in temperature of the strip.

Using (7) and (14^), we have for a strip supplied with direct current
/o and alternating current of effective value t, the acoustic amplitude

{ = —._-— -^^-r '-—. sm I pt — tan-^ — I . (15)

Using (7') and (14a) we have for a strip supplied with alternating current
only

t' = '- ,^ cos I 2pi - tan-i -^ I . (15')

These two equations contain no transient terms; they are solutions for
the state of steadily maintained vibrations. The acoustic amplitude {'
(Equation 15') is of double the frequency of the applied alternating
current.

Using either method of actuating the strip, there is a low critical
frequency above which the factor yp (which represents thermal inertia)
is so much greater than j8 (which represents conduction loss, or dissipa-
tion) that the latter can be neglected. (This frequency is in the neigh-
borhood of 100 for platinum strip i micron thick.) Neglecting j8, (15)
can be written

and instead of (15') we have

f ' ^'T— cos {^Pt-j}. {15'a)

In considering how the efficiency of the process depends on the con-
stants of the strip, we note that it is advantageous to make the resistance
R as large as possible, and the heat capacity ay as small as possible.
The advantage of thinness is plain.

In calculating the intensity of a sound wave, or the rate of flow of
energy in the medium it is necessary to know the square of the particle
velocity; and this is, from (15a) (using superimposed direct current)

Similarly from (15'a), for alternating current,

^ 1.2 X lo-F^i*

Na*!^*] THERMOPHONE AS A PRECISION SOURCE OF SOUND. 3 1

These equations enable us to find the strength of the source; and knowing
this, we can calculate the intensity of the sound at any distance from
the source, in the ideal case in which energy is propagated in the form of
spherical waves in a. homogeneous medium.^

Since the dimensions of the source are small compared with the wave-
length of sound, we may consider the strip as equivalent to a small
sphere of the same area (2a) and which produces the same fluid velocity
i at the surface. The velocity potential for the resulting spherical
distribution of sound waves is

4Tr

U'-if)- <■')

in which 2ai^^ is the strength of the source, or maximum rate of
emission of fluid at the source. In order to calculate the intensity of
the sound produced, we make use of the two following equations

W IP
Intensity = - = --^ , (18)

dtp , .

n=-p.^. (19)

in which n is the pressure change at any point in the field, c the velocity
of sound, and po the mean density of the medium. Substituting (17)
in (19) we obtain for n in terms of J,

>mftx

2Tr

and for the intensity, according to equation (i8)

/ 2Cf*

(20)

or finally in terms of the electrical energy used in the strip (direct current
case)

T7 2 X io-^i?*/oVpof

or, for alternating current,

W 6.0 X io-"i?2iVo/

(21)

/ cr7

2^2

(21')

* The solution here given for intensity in the case of ideal spherical distribution may easily
be applied to the more practical case in which the small thermophone element is placed
close to an infinite rigid plane wall. In this case, the velocity potential on the thermophone
side of the wall will be twice as great as given by (17) and the intensity four times as great
as given in (ao).

32 H, D, ARNOLD AND I. B. CRANDALL. [

Thus the actual intensity at any point some distance away from a thermo-
phone whose power input is constant should increase with the first power
of the frequency, and decrease with the square of the distance r. It is
independent of a, the area of the strip.

Second Case: Production of Pressure Changes in Small Enclosure. —
Let us assume that the strip is placed in an enclosure the dimensions of
which are small compared with the acoustic wave-length, and further
that the shortest distance from the strip to the boundary is large compared
to the wave-length of the heat wave originating at the surface of the
strip. These conditions are readily satisfied for all ordinary acoustic
frequencies. If the temperature variation of the strip is given by

T' sin (at

the temperature variation at any near-by point in the enclosure is

r,' = r'c— ' sin {o3t ± ax). (8)

We can consider that both sides of the strip, each of area a, give rise
jointly to the temperature wave; also that this temperature wave travels
a mean distance x before striking boundary defined by the equation

- Fo
X = — ,
2a

where Fo is the volume of the enclosure. The alternating temperature
averaged over the whole enclosure is* then

2a r* 2aT r*

bT = — \ TJdx = -Tjr- I £-^ sin (w/ - ax)dx. (22)

The thermal conductivity of the gaseous medium varies as the square
root of the absolute temperature, while the specific heat per unit volume
is practically constant at constant volume, so that the diflfusivity is

In terms of Xo, the diflfusivity at 0° Centigrade, Bi is the absolute tem-
perature of the gas near the element, this being approximately the same
as the temperature of the element itself.
We then have

As a varies only as the fourth root of V273» and conditions are easily
arranged so that the temperature of the gas is not excessive, a may be
considered constant in the evaluation of the integral in (i 7) . Integrating,

No*!^'] THERMOPHONE AS A PRECISION SOURCE OF SOUND. 33

— sin (at — cos (at I .

Now Ko is of the order of unity (Kq = 1.5 for hydrogen and .23 for air,
using specific heat at constant volume) so that a (equation 23) is large for
all acoustic frequencies. We may, therefore, neglect ^'^^^^^ and write

and, substituting the value of a from (23)

2aT

6T = -

ljK^^^sin(.t-l). (.4)

If the walls of the boundary are rigid, we have for a perfect contained
gas, 6V = o and the pressure change in terms of temperature change is

if P = total pressure and 6% is the mean temperature of the gas. Sub-
stituting dT from (24) we have for pressure change in the enclosure

^ 2arP^Ko^ei/273 .
n = :;= sm

(<./-j), (25)

etVo^(a ^ 4

in terms of temperature variation in the strip. When direct current is
used with the A.C. this is. given by (7) ; substituting this expression for
T' and dropping the dissipation factor fi, we have, (w = p)

.oS6RI^P^Ko^ej273 i 3tV , ..

and when the strip is actuated only by alternating current, we have
from (25) and (7')i dropping |8 as before, and noting that « = 2p,

° W^^^ C0S(2/,/--). (26)

In (26') / is the frequency of the alternating current and half the acoustic
frequency.

Equations (26) and (26') are in the most convenient form^for calcu-
lating the stress exerted on any part of the boundary, which may be
the exposed face of a sound detecting mechanism, as for example the ear.
The intensity of the sound produced in the enclosure can easily be com-

34 H. D, ARNOLD AND /. B, CRANDALL. [

puted from the usual equations

W IP

— = hp^s^ = i — . (27)

t PqP

in which s = maximum condensation (n/P), n = maximum pressure
change, po = mean density, and c = velocity of sound in medium.
Substituting the value of n from (26) in (27), the intensity is, in the
case of direct current operation

W 3.7 X lO-^moVP'Ko^JjjTn (28)

and in the case of alternating current only, from (26')

W ^ I.I X io-^m'P'Ko^/e,/273

It is seen from these equations that the intensity in this case is in-
versely proportional to the cube of the frequency. The temperature d
has been retained in equations (28) and (280t and the calculation has
been carried through to a determination of the intensity; but there is
not much difference between equations (21), (21') which deal with the
intensity in the first case, and (28), and (28') which deal with the intensity
in the second case, except the frequency-variation law.

In all cases the temperatures of gas and of strip must be taken into
account; and in most cases it is possible to arrange experimental work
and calculation so that this can be done in a very simple way.

Experimental Tests.

The first test that was made was a rough verification of equations
(15a) and (26) to see if the computed effect was of the right order of
magnitude. The method used consisted in setting the thermophone
and an electro-mechanical source (ordinary telephone receiver) for equal
intensity at the same pitch, and measuring the electrical input into each
instrument. The setting for equal intensity was made with the unaided
ear, for simple experiments have shown that the ear judges equality
between two tones of the same pitch to within 4 or 5 per cent.^ The
telephone receiver had previously been calibrated as a sound generator
by measuring the motion of the diaphragm with a microscope when a
known value of alternating current was sent through it. In the case
of the vibrating telephone diaphragm, the motion of the diaphragm is
greatest near the center, falling off to zero at the edge. The law of

^ Or to one per cent, under favorable conditions. The ear seems to be about as good in
these measurements as the eye is in the analogous photometrical case.

No*!^] THERMOPHONE AS A PRECISION SOURCE OF SOUND. 35

distribution of amplitude over the diaphragm is, for small vibrations
(at the particular "frequency used), such that the bowed diaphragm
may be considered from the standpoint of air displacement as replaced
by a piston whose area is 0.306 that of the diaphragm, and which moves
back and forth with an amplitude equal to the amplitude of the diaphragm
at the center.

The data of this experiment were:

Frequency, 800.
Constants of telephone receiver:

Area of diaphragm, 18.3 sq. cm.

Effective area, 5.5 sq. cm.

800-cycle current, 1.7 X io~* amp.

Amplitude at center of diaphragm 1.85 X io~* cm.
Constants of thermophone element:

Material, platinum, of thickness 7 X lO""* cm.

Area a = 0.8 sq. cm.

Effective area 2a = 1.6 sq. cm.

y = (thickness times specific heat per unit volume) = 5 X io~*

Resistance i.o ohm

Direct current /© = i .2 amperes.

800-cycle current = 5.6 X lO"^ amp.
The amplitude (fnuo) >s computed from (15a) corrected for tempera-
ture as per (14) :

6.4 X iQ-^RIoi' ^273

Allowing for a temperature of about 150® centigrade (6 = 423), we
compute

fniM = 4-2 X io-« cm.

In comparing the acoustic outputs from these two sources, we shall
assume that they are two pistons which communicate their amplitudes
of motion to the adjacent medium. The strength of each source should
be proportional (at fixed frequency) to the area of the piston times the
amplitude of its motion. In the case of the telephone receiver, this
quantity is 5.5 X 1.85 X io~' = 1.02 X io~*^ cm.'; and in the case of
the thermophone element, 1.6 X 4.2 X io~' = 0.67 X io~* cm.' In
these experiments the thermophone element was fitted into a receiver
case, similar to that of the telephone receiver, and both instruments
were held loosely to the ear. Assuming them to be tightly held it would
be more correct to compute, instead of displacement, the relative pressure
changes in the enclosed volume of air, (Vq) in order to compare the two

H. D, ARNOLD AND /. B. CRANDALL,

[

i^^- coN$i:x^'

sources. In the case of the telephone receiver the pressure change
would be

n = 1.02 X 10^^

P V,

and for the thermophone, using equation (26)

n _ 0.89 X 10-*

The agreement between the two values, computed in either way is fairly
good, considering the number of factors that have to be taken into
account in making the comparison.

A second experimental test was made for the purpose of verifying the
TMcoiimeAL neuTive ^timn intensity-frequency relation given

in equation (28). Ear comparison
of intensities was again resorted to,
the energy from the strip conduc-
tor being compared with that from
a special telephone receiver at vari-
ous frequencies. (The dynamical
characteristics of the telephone re-
ceiver had been roughly determined
so that it was possible to regulate
it for equal acoustic output at vari-
ous frequencies by adjusting the
alternating current input.) The
A.C. power input i^R in the strip
was measured for equal intensity at
several frequencies, and the results
are shown in Fig. 3.

The points represent the relative
intensity at different frequencies
for equal A.C. power input, and
are proportional to the reciprocal
of the power input for equal in-

1000

100

iNTtHsmr
ARBmumruNirs

(U».KAU)
10

Fig. 3.
Intensity-Frequency Relation in Enclosure.

tensity at each frequency. The curve represents the theoretical decrease
in intensity according to the cube of the frequency, and the general re-
sult is a confirmation of this relation.

The writers are indebted to Mr. E. C. Wente of this laboratory for
an experimental method and data which afford a much more accurate
and satisfactory test of the theory than the two experiments given
above.^ The thermophone element was placed in an enclosure whose

» This experiment was carried out by Mr. Wente in connection with work on the theory
and calibration of a new phonometer which is reported on in the paper immediately following.

No. X. J

THERMOPHONE AS A PRECISION SOURCE OF SOUND,

volume Vo was about 45 cubic centimeters; one of the walls of which
consisted in a phonometer or pressure-measuring instrument as shown
in Fig. 4a, This wall yielded so little that the experiment can be con-
sidered as carried out rigorously under constant volume. The pressure
change in this case, if only alternating current is used to actuate the
strip, is given by equation (26'). The experiments were made at a
frequency of 20 cycles, the (platinum) strip being made sufficiently
heavy to give a large value of ther-

Wz,//>mMt/^

PHONC

^jai^A

mmhim

-n^cc

v///^/m^^

Mton

Fig. 46.

mal inertia ^p so that the dissipa-
tion term j8 could be neglected.
In order to eliminate an absolute
calibration of the phonometer, a
second experiment was made, using
the piston apparatus shown in Fig.
4i, at the same frequency. The
maximum pressure change n as produced by the piston is easily calcu-
lated from mechanical considerations, and the comparison is easily made.

When the piston apparatus was used, the ratio of phonometer reading
to calculated pressure increase was 2.02 arbitrary units; and when the
strip conductor was used, the ratio of the phonometer reading to pressure
change as calculated from (26') was 1.92 on the same scale. This
confirmation to within 5 per cent., was the best we have had of the
theory given in this paper.

The results obtained with platinum show that good quantitative
work can be done with the thermophone when this material is used for
the element. However, it is possible to obtain other materials, such as
gold leaf, which are much thinner than bolometer platinum — and which
are therefore very useful in cases where higher efficiency is needed.
Caution should be used in applying the theoretical formulae to elements
of gold leaf since the heat capacity of gold leaf seems to be very different
in different samples. Any such variations, due perhaps to absorbed
gases, may be cared for (as shown by E. C. Wente in the following
paper) if a check can be made against a platinum element in the same
atmosphere. The correction factor thus obtained should hold for all
frequencies so long as the gold foil is not unduly heated.

Comparative Value of the Thermophone as a Laboratory Source

OF Sound.

With regard to efficiency the thermophone compares favorably with
electromagnetic and electrostatic devices except in the vicinity of their
natural frequencies. In certain work it is essential that the response

38 H. D. ARNOLD AND I. B, CRANDALL, [^SS

should be as nearly uniform as possible over a wide range of frequencies
and that the relative response should be easily determinable. For such
work the advantages of the thermophone are evident, for while its
response diminishes with increasing frequency the law of variation is
simple. When sound of indeterminate loudness and of one frequency
only is desired the volume obtainable from the thermophone does not
compare favorably with that from resonant mechanical devices.

The thermophone is particularly adapted to laboratory purposes be-
cause it requires no adjustment. It is extremely simple in structure
and the units are readily reproducible. The determination of the
acoustic effect of the thermophone depends principally upon the thermal
properties of materials and is remarkably simple as compared with
corresponding determinations for resonant apparatus, which usually
involve motions of complicated mechanical systems. In addition, the
response of the thermophone is uniform through indefinite periods of
time and is not subject to the trouble of accidental detuning, which so
often occurs in resonant apparatus.

Possibly even more important than the ease of determination of the
sound effects in the air close to the element is the fact that these sound
effects cannot react appreciably upon the source of energy whence they
arise. Whenever a vibratory system is used it is always subject to
reactions which may present serious complications. The thermophone
seems the nearest equivalent to an ideal piston source at present obtain-
able.

Various modifications of size, shape and electrical resistance of the
thin conductor employed may be necessary in experimental work.
These need change the theory given in no essential way. On account
of its simplicity from theoretical and practical points of view we believe
that the thermophone in conjunction with a suitable supply of alter-
nating current will be of material value as a precision source of sound.

Summary.

1. A description of a simple thermophone structure is given together
with the theory of its operation.

2. An account is given of experimental tests the results of which are
substantially in accord with the theory.

3. The thermophone is adapted to two classes of service (a) as a
precision source of sound at any frequency (6) as a source of sound of
known relative loudness at different frequencies throughout the acoustic
range.

Research Laboratory of the American

Telephone and Telegraph Co. and Western Electric Company, Inc.

Na*!^] ABSOLUTE MEASUREMENT OF SOUND INTENSITY. 39

A CONDENSER TRANSMITTER AS A UNIFORMLY SENSI-
TIVE INSTRUMENT FOR THE ABSOLUTE MEASURE-
MENT OF SOUND INTENSITY.

By E. C. Wbntb.

THE various methods that have been used with more or less success
for measuring the intensity of sound may be divided into five
general classes: observation of the variation in index of refraction of
the air by an optical interference method; measurement of the static
pressure exerted on a reflecting wall; the use of a Rayleigh disc with a
resonator; methods in which the motion of a diaphragm is observed
by an optical method ; the use of some type of telephone transmitter in
connection with auxiliary electrical apparatus. The apparatus of either
of the first two methods is non-resonant and hence the sensitiveness is
fairly uniform over a wide range of frequencies. These methods are
not sufficiently sensitive, however, to be of use in general acoustic meas-
urements. On the other hand, instruments of the last three classes
possess a natural frequency and are consequently very efficient in the
resonance region. However, in the neighborhood of the resonant fre-
quency the efficiency varies greatly with the pitch of the tone. It is
possible to use a Rayleigh disc without a resonator, but its sensitiveness
in that case is so low that it is of little practical value.

Because of the recent advances in the development of distortionless
current amplifiers, the last class, in which use is made of some form of
telephone transmitter, seems to offer the greatest possibilities. In the
following pages a transmitter is described which has been calibrated in
absolute terms for frequencies from o up to 10,000 periods per second
and which has a nearly uniform sensibility over this range. The appa-
ratus is easily portable, and possesses no delicate parts, so that, when
once adjusted, it will remain so for a long period of time.

Except in cases where measurements are made with a single, continu-
ous tone, it is desirable that the instrument for measuring the intensity
of sound should have approximately the same sensibility over the entire
range of frequencies used. This is especially important if the sound under
investigation has a complex wave form. To avoid any great variation
with frequency in the sensibility of a phonometer employing a vibrating

40 E, C. WENTE. [;

system, it is necessary that the natural frequency lie outside the range
of frequencies of the tones to be measured. Even if the natural frequency
be compensated for in other ways, small variations in the constants of
the instruments, which are always likely to occur, may change conditions
appreciably at this frequency. It is pretty well recognized that for
several reasons the natural frequency should lie above rather than
below the acoustic range. If the instrument is to be used in studying
speech, the natural frequency must indeed be very high. The upper
limit of the frequencies occurring in speech is not definitely known, but
it probably does not come below 8,000 periods a second. Titchener^
found that? if a Galton whistle was set so as to give a frequency of 8,500,
the tone emitted could not be distinguished from an ordinary hiss.

An instrument that is to be used in studying speech should have high
damping as well as a high natural frequency in order to reduce distortion
due to transients. This is not so important if the natural frequency
lies beyond the acoustic range, but nevertheless is desirable even in this
case. Aperiodic damping is the best condition, but it is in general hard
to obtain when the natural frequency is very high.

It seems best in this paper to give a rather complete treatment of the
condenser instrument; for the sake of clearness, however, breaking up
the matter into a number of sections as follows:

1. Theory of the Operation of an Electrostatic Transmitter.

2. General Features of the Design of the Instrument.

3. Deflection of the Diaphragm under a Static Force. Measurement

of Tension and Airgap.

4. Sensitiveness of the Transmitter at Lx)w Frequencies.

5. Sensitiveness at Higher Frequencies Determined by the Use of a

Thermophone.

6. Natural Frequency and Damping of the Diaphragm.

7. Possibilities of Tuning.

8. Characteristic Features of the Instrument.

9. The Electrostatic Instrument used as a Standard Source of Sound.
10. Summary.

Some of these sections deal with theory and some with experimental
work as need arises, the general aim being to put in proper order the
material necessary for a full account of the condenser instrument.

I. Theory of the Operation of an Electro- static Transmitter.

The device to be described is a condenser transmitter, the capacity
of which follows very closely the pressure variations in the sound waves.
The use of such a device as a transmitter is not a new idea; in fact it

* Proc. Am. Phil. Soc., 53, p. 323.

VouX.l
NO.Z. J

ABSOLUTE MEASUREMENT OF SOUND INTENSITY.

was suggested almost as early as that of the corresponding electro-
magnetic instrument.^ However, before good current amplifiers were
available little or no use was made of electrostatic transmitters because
of their comparatively low efficiency.

A simple circuit that may be used with such a transmitter is shown
in Fig. I. When the capacity of the transmitter is varied, there will be
a corresponding drop of potential across R, which may be measured
with an A.C. voltmeter or some other suitable device.

Cf^C 9wa)t

h-^-

Fig. 1.

In order to get a quantitative expression for the magnitude of this
voltage let us assume that the capacity at any instant is given by

C = Co + Ci sin (atf

in which w = 2t X frequency. For the circuit shown in Fig. i

E- Ri ^ ^fidt.

(I)

By differentiation and substitution we obtain

di
(Co + Ci sin o)t)R J. + (i + RCi(a cos (at)i — ECiU) cos w/ = o. (2)

In order to evaluate this equation let us assume as a solution

i = 2)/n sin (nw/ + <t>n).

Substituting this value of i in (2) and determining the coefficients,
we have

•

» =

ECi

>l(co«)

(sin wt + <pi)

+ Ii*

ECi*R

Cc*

N l(c» J

^][(0 + ^

+ terms of higher order in Ci/C©,

]

sin (2aj/ + ^ — ^)

(3)

in which

<tn = tan"** 7; — - and <p2 = tan~^

CocoR

*La Lumiere Electrique, Vol. 3, p. 286, 1881.

2Co(aR '

etc.

42 E. C. WENTE. [fSSS

For the best efficiency R should be made large in comparison with
i/Cow. In this case, the expression for the voltage e becomes

e = Ri = -pr sin (w/ + <pi) — -77^ sin (2w/ + <pi — <pi) + " -.

Co 2Co

From this equation we see that in order to get a voltage of pure sine
wave form for a harmonic variation of capacity, Ci must be small in
comparison with 2 Co. This condition is satisfied as long as the A.C.
voltage is small compared with £.

Retaining only the first term in (3) we have

^ = -Ri = , sin (w/ + (pi), , .

This equation shows that, so far as its operation in the circuit is con-
cerned, the transmitter may be considered an alternating current gen-
erator giving an open circuit voltage £(Ci/Co) sin (w/ + ^1) and having
an internal impedance i/Cow. It can also be shown that the trans-
mitter can be regarded from this point of view if R is replaced by a leaky
condenser or an inductance, so that this result may be said to be true
in general.

2. General Features in the Design of the Instrument.

The general construction of the transmitter is shown in Fig. 2, from
which the principal features are evident. The diaphragm is made of
steel, 0.007 cm. in thickness, and is stretched nearly to its elastic
limit. The condenser is formed by the plate B and the diaphragm.
Since the diaphragm motion is greatest near the center, the voltage
generated, which is proportional to Ci/Cq, will be greatest if the plate
is small. On the other hand, since Co is proportional to the size of the
plate, it cannot be made too small or the internal impedance of the trans-
mitter will be too great. Therefore from the standpoint of efficiency,
a compromise has to be made in determining the area of the plate.
However, if it is made much smaller than the diaphragm, the natural
frequency of the vibrating system will be decreased, as is explained
below. On the basis of these factors the size of the plate indicated was
judged to be about the best for the transmitter.

After some experiments with various dielectrics between the plate
and the diaphragm it was concluded that air was most suitable. The
dielectric constant of air is not so high as that of some other materials,
but its insulating properties are better. However, the principal ad-
vantage of using air is, that it has a high minimum value of sparking

NcTi. "J ABSOLUTE MEASUREMENT OF SOUND INTENSITY. 43

potential which lies in the neighborhood of 400 volts, below which

there is no appreciable conduction. When E is less than this voltage,

the air gap may be decreased without decreasing £, so that the efficiency

of the instrument is limited practically only by the fact that when the

gap is decreased below a certain value, the electrostatic force between

the plate and diaphragm deflects the latter sufficiently to short circuit

the condenser. When a potential difference of 320 volts was applied

to the transmitter shown in Fig. 4, no appreciable current flowed across

the air gap, certainly not more

than io~* amperes. The fact

that the air has such a high A

minimum sparking potential is Punc,

one of the principal reasons why

it is possible to design a sue- -•5*»«i

cessful condenser transmitter of

the type shown in Fig. 2. iks« »te»Mb^

A word may be said in regard
to the method of adjusting the «u««r

transmitter so as to obtain a
small uniform air gap. The sur-
face of part A, next to the dia-
phragm, was ground plane before
assembling. Small irregularities
in the surface of the diaphragm ^^-

facing the plate were removed by Sectional drawinj of tranBmitter.

grinding with flne carborundum.

Parts B, C and D were first assembled without the mica washer. The
face of the plate and the ends of part C were then ground to the same
level. Finally the mica washer was inserted between C and D and the
whole apparatus assembled as shown. The mira may be split into washers
of very even thickness, and a uniform air gap so obtained. The dia-
phragm is clamped between parts A and C, and is thus held in a true
plane. In assembling the parts, the greatest care must be taken that
no dust is caught between the plate and the diaphragm, for the insula-
tion may be considerably reduced by the presence of any small particles
in the gap.

Part C does not fit so perfectly against the diaphragm that the space
surrounding the plate is shut off completely from the outside air.
Changes in tem[>erature and atmospheric pressure will therefore not
affect the equilibrium position of the diaphragm.

The instrument used in these experiments was constructed just as

£. C. WENTE.

Sbcond
Sbsiis.

shown in Fig. 2. It is evident from this figure that the diaphragm may
be brought into contact with the plate if a mechanical pressure is acci-
dentally exerted on the diaphragm. This will cause a spark to pass,
if the transmitter was previously charged. In order to avoid damaging
the metal surfaces in this way it may be advisable to glue to the face
of the plate, ^4, a very thin layer of mica of uniform thickness, while still
retaining an air-gap sufficient to allow free motion of the diaphragm.

3. Deflection of the diaphragm under a Static Force; Measure-
ment OF Tension and Air Gap.

It is not difficult to calculate the sensitiveness of the transmitter for
low frequencies from the dimensions of its various parts, provided the
magnitude of the deflection of every point of the diaphragm produced
by a given static force is known. Since the diaphragm is made of very
thin material and the tension is high, we may expect the diaphragm to
behave very much as an ideal membrane, at least for frequencies near
zero. In order to determine how closely this condition is approximated
the following experiment was carried out.

When a static potential is applied between the plate and the diaphragm,
the latter is deflected by the electrostatic force. The deflection produced

""^

».

Dfi

TAH

ROM

cei

ITRC

ori

XUsi

Fig. 3.

Fig. 4.

in this way by a known potential was measured by a device very similar
to that used by Prof. D. C. Miller in his phonodeik.^ By this arrange-
ment the deflection of the diaphragm was magnified 30,000 times. The
mean values of the deflections produced at various points along eight
evenly spaced radii when a potential of 320 volts was applied are shown
in Fig. 3. Points of equal displacement of the diaphragm are plotted
in Fig. 4. The fact that the curves drawn through these points are

1 D, C. Miller, Science of Musical Sounds, p. 79.

VOL.X.

.^•]

ABSOLUTE MEASUREMENT OF SOUND INTENSITY.

practically circles shows that the tension of the diaphragm was very
nearly the same in all directions.

The distance between the plate and diaphragm was also measured
with this apparatus by applying a mechanical force until the diaphragm
touched the plate. The value obtained in this way was 2.20 X lO"* cm.
The capacity of the transmitter was measured on a capacity bridge and
found to be 335 X io~" farads, from which the computed value of the
air gap is 2.25 X io~* cm. The mean of the values obtained in these
two ways is 2.22 X io~* cms.

In order to determine how closely the diaphragm approximates an
ideal membrane, we may calculate the form that the latter would have
assimied under the conditions of the preceding experiment.

Fig. 5.

Referring to Fig. 5, if V is the potential between the plate and the
diaphragm, and T, the tension of the membrane, we have

[

(Pw . I dwl , F*

This relation holds from r = o to r = J?. Let

A =

7»
8irT

and X = log r, then since (w — w^) = (Fg — y)

cPy €^

dx^"^ y'

or, since (w — w^)/{wq — w^) is very nearly equal [to {R^ — r^)/r^ and
(wo — tt^n) is small compared with y©,

in which

A = (wo - w^)/yQ.

» Rayleigh, Theory of Sound, II., p. 318.

46 E, C. WENTE,

From this we get

-r^7<*'-'^[--;-;^] + "- <s)

The total force on the diaphragm is
where d\ is defined by the equation

iriP C 2'^rdr 2tC i ,^\ . »^ /

di Jo / yo Jo ' -R^' 3^0

so that

In the region extending from r = J? to r = a,

From this

F= - 2TrT-r •

i4iP(i - ^) , a
Wr = log •= . (6)

* 2^0 -R

From (5) and (6)

This equation gives the form into which the diaphragm will be bent if it
behaves like an ideal membrane. The curve representing this equation
is shown in Fig. 3. The observed points do not lie very far from this
curve. We therefore conclude that the diaphragm behaves sufficiently
like an ideal membrane, so that no great error will be incurred if this
assumption is made in calculating the sensitiveness of the transmitter
for low frequencies.
From equation (7)

-3i^.l(--i)+'<-«">4l- <«

Hence, if the deflection at the center of the diaphragm produced by a
known voltage is measured, the tension may be calculated from (8).
Results obtained in this way for the diaphragm used in these experiments
are tabulated below.

Voi.X.1
Nax. J

ABSOLUTE MEASUREMENT OF SOUND INTENSITY.

Volts.

Deflection («i^ (cm.).

Tension (T)
(dynee)

(cm.).

200
240
280
320

6.0 X 10-*

6.8
12.4
16.9

6.59 XIO^
6.58
6.55
6.55

Mean

6.57X10^

4. Sensitiveness of the Transmitter at Low Frequencies.

Having satisfied ourselves that the diaphragm behaves sufficiently
like a perfect membrane, and having determined the tension and air gap,
we can now proceed to calculate the efficiency of the transmitter for
low frequencies. To do this it is necessary to find the change in capacity
produced by a given pressure on the diaphragm, since by equation (4)
the voltage generated is proportional to Ci/Cq.

Referring to Fig. 5, we see that the capacity is I^/4d if the diaphragm
is not deflected. From the curve of deformation when a potential is
applied (Fig. 3), it is evident that w^ is very nearly equal to 0.45 Wq,
Hence the air gap at any point is given by

d - Wo +^tt;of*,

and since the surface of the diaphragm deviates but little from a plane
area, the normal capacity to the first approximation is

2irrdr

Co =

4T yd-Wo + '-^-r^

\ 4(PL^ ■*■ d 2d \ 4d"

(9)

'"Jo ~A

in which d' may be called the effective air gap.

If a pressure, P, uniform all over the diaphragm produces a deflection,
u, the capacity of the condenser will have been changed by the amount

2irrdr

-. ^^- ^^°^

The quantity in brackets of equation (9) does not differ greatly from
unity in any practical case, so that no great error will be incurred if we
set y in (10) equal to the constant value d'. Since

« = ^ (a' - f*)/

equation (16) may be written

<:i =

P C"

Z2Td'

(II)

> Lamb. Dynamical Theory of Sound, p. 150.

E, C. WENTE.

fSBCOMD
LS»1I»S.

Ci IS the change in capacity produced by a static pressure, P; but this
diflfers very little from the maximum value of the alternating capacity
resulting from a pressure, P sin «/, provided «/2t is small compared
with the natural frequency of the diaphragm.

Having determined Ci per unit value of P from equation (ii), and Co
from (9), we may calculate CJCq and hence the sensitiveness, t. e., the
volts per unit pressure. In practically all the experiments that have
been made with the electrostatic transmitter, the D.C. voltage was 321.
Under this condition we obtain 315 E.S.U. for C© from (9) and 1.96 X lO"*
E.S.U. per dyne per sq. cm. for Ci/P from (11). Hence we have for the
sensitiveness

EC\ _ 1.96 X 10"* X 321 _ 2.00 X 10"* volts
PCo " 315 dynes per sq* cm. '

In order to check this value directly by experiment, the apparatus
diagrammatically shown in Fig. 6 was constructed. A receptacle was

P»5T0N

OU-
PHRAQM

ref-VWOL

TRAHSHnrCR

MOTOR

Fig. 6.

placed over the diaphragm as shown in the figure, thus forming an air-
tight enclosure. Connected to this was a cylinder containing a piston.
The connecting rod was long compared with the stroke of the piston so
that with the motor running, the piston was given practically a simple
harmonic motion. The fly wheel was fairly heavy and the connecting
rod was made of stiff tubing, so that but little vibration was noticeable
even when the motor ran at the highest speed.
The pressure variation is given by

6V
6P = i^^P-y'

in which 5T^ is one half the total piston displacement and P is the maxi-
mum value of the alternating pressure.

V = 45.2 c.c. (volume of chamber)

dV =

0.68 X 0.418

= .142 c.c.

VoL.X.1
Nai. J

Hence

ABSOLUTE MEASUREMENT OF SOUND INTENSITY,

1.42

5P = 1.4 X 10* X = 4,400 dynes per cm.*

45-2

The root mean square value of the pressure is

4400

^/2

3,120 dynes per cm.*.

The circuit used in this test is shown in Fig. 7. The electrostatic
voltmeter had a very small capacity, giving it at low frequency an
impedance large compared with the 80 megohm resistance in shunt.

nnrw

CLZCTROSTATIC
VOOHCTCR

320 >fQLTS

Fig. 7.

We may then calculate the open circuit voltage given by the transmitter
from the voltmeter reading and the constants of the circuit, remembering
that the transmitter may be regarded as a generator having an internal
impedance i/Co«. The following values were obtained in this way.

Motor SpMd, R.P.lf .

Prcqn«acy (P.P.8.).

Voltmeter Readiof
(Volte).

Open Circuit Volte.

1239

1074

950

824

584

20.7
17.9
15.8
13.75
9.75

5.31
5.31
5.31
5.20
4.92

6.22
6.27
6.33
6.29
6.27

Mean

6.28

We therefore have for the sensitiveness,

6.28 . Volts

2.02 X 10"

3120 dynes per sq. cm. *

This value is in very close agreement with that given before, so that
we may consider 2.00 X lo*^ volts per dyne as a reasonably correct
value for the sensitiveness at low frequencies.

5. Sensitiveness at Higher Frequencies as Determined by the

Use of a Thermophone.

By the methods just described the values of sensitiveness may be
determined for very low frequencies only. In order to measure the

E, C. WENTE.

rSBCOMD

Capillary

FOIL
DlAPtlRAGH

Block or lcao

Fig. 8.

sensitiveness at higher frequencies and also to get an idea of the natural
frequency and damping of the vibrating system, use was made of the
principle involved in the action of the thermophone as described by

Arnold and Crandall.^ A block
of lead about 1.5 inches thick
was placed against the face of
the transmitter so as to form
a cylindrical enclosure in front
of the diaphragm, i^ inches in
diameter and ^ inch long. The
general arrangement is shown in
Fig. 8. All crevices were sealed
up so that the only openings to
the cavity were two capillary
tubes several inches long and of
about o.oi cm. bore. Two strips
of gold foil were mounted sym-
metrically inside of this enclos-
ure, the ends being clamped be-
tween small brass blocks. The supports were arranged in such a way
that a current could be passed through the two strips in series. The con-
nection between them was in electrical connection with the diaphragm.
In the paper just cited it is shown that within an air-tight enclosure

— / $1 \i/4
.oio6Ri^Ps/Ko(-t-)

^^= .W3/^ ' (^^)

in which
6P = maximum value of the alternating pressure within the enclosure.
P = normal pressure within the enclosure.
R = resistance of the foil.

i = r.m.s. value of the alternating current passing through the foil.
2^0 = diflfusivity at o® C. of the gas within the enclosure.
^1 = mean absolute temperature of the foil.
02 — mean absolute temperature of the gas.
7 = heat capacity per unit area of the foil.
Vo = volume of the enclosure.
/ = frequency of the alternating current.
Equation (12) may be used for calculating the pressure variation pro-
vided the wave-length of sound is large compared with the dimensions
of the enclosure. The velocity of sound in hydrogen is about four times

» Physical Review (Preceding Paper).

Voi.X.1
Nax. J

ABSOLUTE MEASUREMENT OF SOUND INTENSITY.

as great as in air; hence formula (12) holds for frequencies almost four
times as high, when the enclosure is filled with hydrogen instead of air.
Also, the diffusivity, Ko, is about six times as large for hydrogen as for
air, so that greater pressure variation is obtained with the former.
For these reasons, hydrogen was passed in a continuous stream through
the enclosure by way of the capillary tubes, at a rate sufficiently slow
to prevent any appreciable increase of the steady pressure above that
of the atmosphere. The hydrogen was obtained from a Kipp generator
and then passed through a solution of potassium permanganate and a
dr3ang tube containing phosphorus pentoxide.

In order to get the open circuit electromotive force of the transmitter,
the circuit was arranged as in Fig. 9. The two resonant circuits were so

ron.

QSOLLATOR

Ctooujrot

-WWWV

THCRMO-

c-iM0 9MccAf>Mcrrr

MTWWMITTW

Fig. 9.

adjusted as to prevent current of the same frequency as that given by
the oscillator from passing through the galvanometer. If a pure sine
wave current passes through the foil, the pressure variation in the
enclosure is of pure sine wave form and of double frequency. However,
if there is any second harmonic present in the current, there will also be
a component of the pressure variation of single frequency.^ Putting
in the resonant circuits eliminates this component from the measure-
ments. The general procedure in making a measurement was as follows.
The double-throw switch was first put in position i, and the foil current
and galvanometer current read. The switch was then thrown in posi-
tion 2; R and r were then adjusted until the galvanometer read approxi-
mately the same as before. From the readings ol At and the values of
Ru Ri and r, the voltage drop across r may be calculated. The open
circuit voltage of the transmitter is then obtained by multiplying this
voltage drop by the ratio of the galvanometer readings. That this gives
us the open circuit voltage, follows from the fact that the transmitter

> Arnold & Crandall, loc. cit.

E. C. WENTE.

[Sboomd
Sbsibs.

behaves as a generator having an internal impedance i/Co(a, Oscillator
No. 2, of course, is set at double the frequency of Oscillator No. i.

The current passed through the gold foil was about 0.5 ampere at all
frequencies. Resistance measurements showed that with this current
density, the foil was not heated more than 10° C. above the room
temperature. The values of the quantities entering into the formula
(12) for this experiment were as follows:

J? = 4.18 ohms.

P = 10* dynes/cm*.
Ko = 1.48 C.G.S. units.

Ot = 295^

Bi = 305^

7 = 4.15 X io~* calories per sq. cm.
Thickness of gold foil = 7 X io~* cm.
Width of each strip = i cm.
Length of each strip = 2.6 cm.
Substituting these values in equation (12), we have for the root mean
^uare value of the alternating pressure

2.59 X io'tjj; dynes per sq. cm.

Dividing the measured open circuit voltage by this value should give
us the volts per unit pressure for all frequencies within certain limits.

Measurements were made in

DVMOflRMCit

jaz2.

ncio

CMJORMMM or OONKNXR ItUMMfVCI

this manner for frequencies from
160 to 18,000 cycles per second.
The general shape of the curve
obtained by plotting these val-
ues is shown in Fig. 10.

The absolute value of the sen-
sitiveness at low frequencies as
determined by this method was
0.121 X 10"* volts per dyne,
which is only about one sixteenth
of that previously obtained by
the piston method and by calculation from the dimensions of the instru-
ment. In order to make further tests within the range of frequencies
from 20 to 160 cycles, the gold was replaced by platinum foil, 4.42X10"^
cm. thick, and measurements were made as before. However, the size of
the enclosure was increased in order to meet the conditions assumed in
the derivation of formula (12), and for the same reason air was used
instead of hydrogen.

Fig. 10.

Na*if*] ABSOLUTE MEASUREMENT OF SOUND INTENSITY. 53

Calculations made in a manner similar to that when gold foil was used
gave a value of 1.93 X lo"* volts per dyne per sq. cm. for the sensitive-
ness at low frequencies. This is in fair agreement with the value
2.00 X io~* obtained theoretically and with the piston apparatus.

Apparently when gold foil is immersed in hydrogen something takes
place which is not taken account of in equation (12). The gold foil
used was extremely thin (7 X lo"' cm.) and when placed in hydrogen
its specific heat per unit volume was apparently much greater than that
of pure gold assumed in the calculations. On account of this discrepancy
the gold leaf could not be relied upon for an absolute calibration, but it
seemed reasonable to assume that the ratio between the true pressure
and that calculated was independent of the frequency, so that a true
relative calibration for different frequencies could be obtained. To get
the absolute value of the efficiency at all frequencies, the values calcu-
lated from the readings on the gold foil were multiplied by the factor
2.0/.121 = 16.6. The results so obtained are those plotted in Fig. 10.

6. Natural Frequency and Damping of the Diaphragm.

It is thought that this curve (Fig. 10) may be relied upon to give the
sensitiveness in absolute value for frequencies up to 10,000 cycles.
Above this frequency the wave-length of sound approaches the diameter
of the cylindrical enclosure. The wave-length in hydrogen at 10,000
cycles is 13 cm. whereas the greatest distance from boundary to boundary
of the enclosure is 4.4 cm. Although the absolute values of the sensitive-
ness above 10,000 cycles are probably not given by the points plotted in
Fig. 10, nevertheless, this curve indicates in a general way the behavior
of the transmitter at high frequencies. The principal peak in this curve
comes at 17,000 cycles, which undoubtedly corresponds to the natural
frequency of the diaphragm. The damping cannot be determined with
any great assurance of accuracy, although the curve as drawn would
indicate a damping factor of the vibrating system of about six or seven
thousand.^

These high values of natural frequency and damping are in a large
measure due to the cushion effect of the air between the plate and the
diaphragm. Free lateral motion of the air is prevented by its viscosity.
This increases the rate of dissipation of energy when the diaphragm is
vibrating and also adds to its elasticity.

To see whether 17,000 cycles is a reasonable value for the natural
frequency we may make an approximate theoretical calculation. When

' The term damping factor as here used may be defined as the reciprocal of the time
required for the amplitude to fall to 1/2. 7 18 of its initial value.

54 £. C. WENTE. [gSSS

the frequency is as high as 17,000 cycles it seems reasonable to assume
that there is practically no lateral motion of the film of air. Let us
further assume that the film of air is compressed and rarified adiabatically
by the motion of the diaphragm and also that the plate is of the same
size as the diaphragm. This latter condition is not quite satisfied in
the case of the electro-static transmitter but no great error is introduced
by this assumption, since the motion near the edge of the diaphragm is
small. Under these conditions if d is the length of the air gap, P, the
atmospheric pressure, p, the mass per unit area of the diaphragm, and T,
the tension, the equation of motion of the diaphragm becomes:

^ d? " [dr^'^'rdr) d~ *

or since w varies as c^*'.

The solution of (13), consistent with the boundary conditions, is

w = Mir),

in which

7 ^ fp«* __ 14^
' ST Td '

The boundary conditions require that Jo(la) = o. The lowest root of
this equation is 2.4 so that

[p«»-^]^=(24)'

-'^ " 2,r ~ 2ir Nip \ a / \( ■*■ ^^(2.4)^ '

This equation gives the natural frequency of the diaphragm when vibrat-
ing in its fundamental mode.

For the transmitter used in the preceding tests —

^ = 6-57 X 10^ dynes per cm.
P = '05 gn^- P^r sq. cm.
a = 2.18 cm.
P — 10* dynes per sq. cm.

d = 2.22 X 10"* cm.
Hence

/o = 6,350 ^8.9 = 19,000 P.P.S.

* Rayleigh, Theory of Sound. L., 318.

No*!^*] ABSOLUTE MEASUREMENT OF SOUND INTENSITY. 55

which is slightly higher than the observed value. With the plate re-
moved, the diaphragm would have a natural frequency of 6,350. This
shows that the film of air between the plate and the diaphragm increases
the elastic factor many times. It is due entirely to this fact that it
has been possible to obtain natural frequencies above 10,000 without
making the diaphragm exceptionally small.

We may satisfy ourselves that the maximum point in the efficiency
curve is not due to resonance in the cylindrical enclosure by calculating
its resonant frequencies. These frequencies are determined by the
equation

in which

Jn'i/K^ - P't^I"^ R) = .0^

a -^

a « velocity of sound.

/ = length of cylinder.

R =s radius of cylinder.

^ = an integer.

Since the foil was placed symmetrically in the enclosure, only the sym-
metrical modes of vibration need be considered, in which case « = o.

The first root of the equation Jf!{Z) = o is 3.83. For the lowest
resonant frequency, ^ =» o, so that we have

. fl 3.83

In this problem

a = 127,000 cm./sec. (velocity of soimd in hydrogen).

R = 2.18 cm.
hence

/o = 35,500 cycles per second.

which is very much above the frequencies covered in the calibration.

If the enclosure is filled with air instead of hydrogen, the first resonant
frequency comes at about one fourth of 35,500 or 9,000 p.p.s. A series
of measurements were made with the circuit arranged as in Fig. 9, and
air instead of hydrogen surrounding the gold foil. Points were calcu-
lated and plotted ; the curve so obtained showed a sharp resonant point
at 9,600 but none below. This may be taken as further evidence that
the maximum point in Fig. 10 is not due to any resonance in the enclosure
and so corresponds to the natural frequency of the diaphragm.

» Rayleigh, Theory of Sound. II, 300.

56 E. C, WENTE. ^^^S.

There is an irregularity in the calibration at about 3,500 periods per
second. This is undoubtedly due to the natural frequency of the back-
piece. At any rate, vibration of the plate would have an effect of this
general character, i. e,, the efficiency would be decreased below, and
increased above, resonance. In a later design, the plate and support
have been made more rigid so as to form practically one solid piece.
It is believed that with the newer model, the irr^;ularity in the curve will
have been eliminated.

This completes the account of the experimental work done in calibrating
the instrument.

In order to obtain some idea of the sensitiveness of the electrostatic
transmitter just described as compared with an electromagnetic instru-
ment, the sensitiveness of the former was compared directly with an
ordinary telephone receiver used as a transmitter, over a considerable
range of frequencies. Except within a hundred cycles of the resonant
frequency of the diaphragm of the receiver the electrostatic transmitter
was found to generate a greater voltage for a given sound intensity.

7. Possibilities of Tuning.

. Since an electrostatic transmitter is equivalent to an alternating current
generator having an internal impedance i/Co«, it is evident that, if in
the circuit shown in Fig. i, the resistance R is replaced by an inductance
L, the voltage e will be a maximum for a frequency of

The sharpness of tuning will of course depend upon the possibility of
getting an inductance with a small resistance. In many problems in
acoustics it is desirable to have a tuned system and in that case it is
also better to have a diaphragm of low natural frequency and damping.

In order to get an expression for the sensitiveness as a function of the
frequency, let us assume that we have a parallel plate condenser, one
of the plates of which is fixed and the other moved perpendicularly to
its own plane by a simple harmonic force. Practically this condition
is approximated by a diaphragm, the center of which is separated a short
distance from a plane plate as is shown in Fig. 2.

Let X = displacement of the diaphragm from its equilibrium position.
d =» air gap, Jissumed large compared with x.

Then

C" CaV ■'"d)-

No^xf] ABSOLUTE MEASUREMENT OF SOUND INTENSITY, 57

The mechanical impedance of the diaphragm is

Zi « r + jym<a - -) ,

where r =* resistance factor,

m = mass factor,

s = elasticity factor,

« = 2t X frequency.

If r = kinetic energy of the entire system,

W = potential energy of the entire system,

2F = rate of dissipation of energy,

then

2r = mi* + Ly^,

2W = sx' +^^(i +2) (Y + y)\

2F = i?3^ + rx\

where Y is the permanent, and y the variable electric charge on'the con-
denser. The equations of motion for the system are

±ieT\^dT dF dW^
dt\dxf dx '^ dx '^ dx '

d_/drv BT dF dW
dtXdy / dy'^ dy'^ by ^^'

If second order quantities are neglected, and also the constant terms,
which affect only the equilibrium position, these equations become

pmx + r« + T ic +

p ' pCf4

pLy + Ry+j^^y + ^^l^o

in which p is written for

solving equations (14) for y and substituting the values,

E = Y/Co, Z, = (r + s/p + pm), Z = (R + i/pCo + RL),

we have

. PE

^ "" pd[{Elpd)* - ZiZ] '

(14)

58 £. C. WENTE, [

PE{R + pL)

e^y{R + pL)

pd[{E/pdy - ZiZ] '

In any practical case {E/pdy is small compared with ZiZ so that we
may write without much error

^ EP(R + pL)
^ ^ pdZiZ '

In order to obtain a large value of e/P, which is a measure of sensitiv-
ness, ZiZ should be made small, i. e., the diaphragm should have a natural
frequency equal to the frequency, «/2t, and the electrical circuit should
be in resonance at the same point.

No extensive measurements have been made with the circuit arranged
in this way, although enough has been done to show that it is feasible
in some cases. The chief difficulty lies in the fact that the transmitter
capacity is so small that the inductance has to be very large to get
resonance for ordinary sound frequencies. This difficulty may be over-
come by shunting the transmitter with a condenser, which of course
reduces the generated voltage.

8. Characteristic Features of the Instrument.

Because of the high internal impedance of the electrostatic trans-
mitter, it is possible to use the instrument efficiently only with high
impedance apparatus, such as an electrostatic voltmeter or a vacuum
tube amplifier. However, this is no special disadvantage if an amplifier is
to be used, because it is not desirable to use a transformer in connection
with an instrument for measuring sound intensities, since the ratio of
transformation of a transformer is not independent of either frequency
or load.

The method for calibration of this instrument as explained in the pre-
ceding pages is rather elaborate and requires considerable care. But
since the efficiency depends primarily on the air gap and tension, it
should not be difficult to make duplicate transmitters to which the same
calibration applies, since the desired values of air gap and tension may
be obtained without great difficulty, the former being tested by measuring
the capacity, and the latter by determining the deflection produced by a
known potential between the plate and diaphragm.

The fact that the sensitiveness of this instrument is independent of
any properties of material, such as magnetization or electrical resistance,
is of considerably advantage. For this not only allows us to make instru-
ments which are almost exact duplicates, and so let the calibration for

Na"i^'] ABSOLUTE MEASUREMENT OF SOUND INTENSITY, 59

one instrument serve for all the rest, but, the calibration is also constant
with the time. The metal parts are of machine steel throughout; from
the construction as shown in Fig. 2, it is therefore evident that tempera-
ture can affect the sensitiveness but little. The tension of the diaphragm
is, of course, not absolutely independent of temperature, nor is the action
of the cushion of air between the plate and the diaphragm independent
of the barometric pressure: but these effects are hardly worth consider-
ing. Being made of heavy material, the transmitter satisfies the require-
ment in the way of ruggedness; having once been adjusted, it should
remain so, even if subjected to considerable rough usage.

The sensitiveness of the transmitter is not absolutely uniform, but
varies only about a hundred per cent, between zero and 10,000 cycles,
as the curve in Fig. 10 shows. This variation is much less than would
be the case with an electromagnetic instrument with a diaphragm having
the same natural frequency and damping. Except for eddy current and
iron losses, the voltage generated by an electromagnetic transmitter is
proportional to the velocity of the diaphragm, whereas that given by
the electrostatic transmitter is proportional to the amplitude. Below
the natural frequency, the variation of velocity with frequency is much
greater than the variation of amplitude since the velocity is proportional
to the product of the frequency and amplitude.

In most problems the transmitter would be used with an amplifier.
Now, the sensitiveness of the transmitter increases, whereas the efficiency
of an amplifier sometimes decreases with the frequency; at any rate,
it is possible to design a circuit for the amplifier, so that the combination
of the two has a constant sensitiveness over a wide range of frequencies.

Since the natural frequency of the transmitter is very high, instan-
taneous records of sound waves obtained in combination with a dis-
tortionless oscillograph would not only give the relative amplitudes of
the different frequencies into which the sound may be analyzed, but also
the phase relations should be practically unchanged for frequencies up
to 10,000 p.p.s.

As yet no instrument is available which will record without distortion
currents of frequencies as high as 10,000 cycles. Only after such an
instrument has been developed will it be possible to get a true record of
consonant sounds. The same is true in regard to the quantitative study
of the quality of musical instruments. However, by using an ordinary
high frequency oscillograph in connection with a condenser transmitter
and amplifier, it should be possible to get curves equal to or better than
any obtained heretofore.

60 £. c. wente. [isss

9. The Electrostatic Instrument Used as a Standard Source of

Sound.

There is of course no theoretical reason why the instrument described
in the preceding pages cannot be used in a reversible manner: that is,
as a source of sound when an alternating voltage is applied between the
plate and the diaphragm. If the instrument is to be used in this way,
it is better to have the plate the same size as the diaphragm, in order to
get the maximum electrostatic force for a given voltage and air gap.
The resulting increase in capacity is in general no disadvantage in this
case. Also for convenience in using the instrument it may be desirable
to have the face of the plate covered with a thin layer of mica.

Because of the simplicity of this type of instrument it is not difiiailt
to calculate the output of sound energy for a given voltage after its
efficiency as a transmitter has been determined. It is evident that the
instrument can be excited in two different ways; (a) the alternating
voltage can be applied alone, and (b) it may be superimposed on a static
potential maintained by a battery in exactly the same way as when the
instrument is used as a transmitter. The main principles underlying
the two kinds of excitation in this case are quite similar to those discussed
by Arnold and Crandall in connection with the excitation of the thermo-
phone by pure A.C. and by A.C. with D.C. superimposed. For this
reason neither type of excitation need be discussed at length; but a brief
treatment of the condenser instrument excited by pure alternating current
will be given.

When a pure alternating voltage is applied, the mean deflection of the
diaphragm will depend on the magnitude of this voltage and the efficiency
may vary somewhat because of the change in mean air gap, and the conse-
quent change in the cushion effect of the air sheet on the motion of the
diaphragm. It is therefore necessary to have curves corresponding to
the curve in Fig. 10 but for a series of applied static potentials. These
are most easily obtained by determining for a number of frequencies the
generated voltage as a function of the static potential when sound of a
fixed intensity falls on the transmitter. It will be found that the alter-
nating voltage generated is so nearly proportional to the static potential
that for most acoustic work this may be assumed to be the case.

When an alternating potential ^2v sin cot is applied to the plates,
the electrostatic force per unit area acting on the diaphragm is

sV^Cl - cos 20)/). (15)

Now refer to Fig. 10, assuming that the curve there shown gives the

No*!^*] ABSOLUTE MEASUREMENT OF SOUND INTENSITY. 6 1

efficiency of the instrument, used as a transmitter^ for an applied static
potential v. If we multiply the ordinate (t. ^., the voltage per unit
pressure) at frequency «/t = / by the quantity

C/o Co

we can obtain (cf. (4)) C/, the change in capacity per unit pressure. JThe
total change in capacity due to the electrostatic force is then, (if C is
the change per unit pressure at zero frequency)

Ci = g^ (C - C/ cos 2«/) (16)

from which we can proceed to calculate the amplitude of motion of the
diaphragm.

It is necessary of course to have a mean value of d, the air gap, but
it is sufficiently accurate to take an arithmetic mean of the values at the
center and at the edge of the diaphragm. The motion at the center is
greater, but the motion near the edges extends over a greater area.

In computing the mean amplitude of the diaphragm we shall introduce
very little error if we take the form of the diaphragm as that of a para-
boloid, tt, the amplitude at any radius, r, is given by the relation already
quoted

«=^(a*-r»), (17)

in which a = the radius of the diaphragm and plate. Equation (ii)
gives the total change in capacity in terms of P and T, that is (since

or, eliminating P/T between (11') and (17) we have, for displacement at
any radial distance r, in terms of maximum capacity change

tt = -^(a'-r*)Ci.

Substituting for Ci the value given in (16) we have

yi — -

« = — (a« - r^){C - Cf cos 2a)/), (18)

in which v is the r.m.s. value of the applied alternating voltage, and C/

62 E. C. WENTE. [SS22?

IS the change in capacity per unit pressure, determined in the manner
described from the calibration curve of the instrument used as a trans-
mitter. Equation (i8) is rigorously true for all frequencies within the
range of calibration, because the quantity C/ is taken from the calibration
curve.

If, however, T is known, we can obtain an approximate value of u
good at low frequencies, without any knowledge of Cu (This is merely
"equilibrium theory" and makes use only of the elastic factor, leaving
the inertia and mechanical resistance of the moving system out of
account). Substituting the value of electrostatic force (15) for P in
(17) we have

v^(a* - f*)

The actual acoustic effect may be determined by the usual methods.
If the diaphragm forms a wall of a small enclosure, the intensity is deter-
mined by the ratio of the volume displaced by the diaphragm as it vibrates
to the volume of the enclosure. In other cases the intensity at a given
point is calculated by determining the velocity potential due to the
motion of the diaphragm.

It has been tacitly assumed that the amplitude of motion of the
diaphragm is small compared with the air gap. This is necessary in
order to get a pure tone when a sine wave E.M.F. is applied. While
the instrument will not take care of a very large amount of energy,
sound of the same order of intensity may be obtained as from an ordinary
telephone receiver without appreciable distortion.

Summary.

1. A description is given of a transmitter of the electrostatic type

which is especially adapted for measurement of sound intensities over
a wide range of frequencies. The instrument is portable and is sufficiently
rugged to retain its calibration.

2. A discussion is given of the necessary auxiliary apparatus and the
precautions necessary for proper use.

3. A theory of the transmitter has been developed by which its opera-
tion can be predicted from a few simple measurements.

4. A description is given of the calibration of such an instrument in
absolute terms over a wide range of frequencies. It is found that its
efficiency may be made practically uniform for frequencies up to 10,000
cycles per second, and the results of the calibration are in agreement
with the theory.

No'if'l ABSOLUTE MEASUREMENT OF SOUND INTENSITY. 63

5. The apparatus when once adjusted may be used for the measure-
ment of the intensities of sound at any frequencies throughout this
wide range without further special adjustment.

6. Due to the uniform response through this wide frequency range it
will be possible to secure correct indications of complex wave forms
and to determine not only the relative intensities of the components
but also their phase differences.

7. When properly calibrated this apparatus can be used as a precision
source of sound.

Rbsbarch Laboratory of thb American

Telbphonb & Tblbgraph Co. and Western Electric Co., Inc.

64 WILLIAM FRANCIS MAGIE, [^mu

THE RELATION OF OSMOTIC PRESSURE TO TEMPERATURE.

By William Francis Magib.
II.

1. Introduction. — ^The present paper is an extension of one published
under the same title in this Review, Vol. XXXV., p. 272, 1912. In
it the formula proposed in the former paper for the relation of the osmotic
pressure to the temperature is deduced from a study of the boiling
point and freezing point cycles, and certain improvements, consequent
upon a more careful consideration of the quantities involved, are intro*
duced. The formula is then tested by comparison with Emden's observa-
tions on vapor pressure.

The formula proposed in the former paper for the osmotic pressure

was

p ^ as \og9 + Q) - a)e + e (i)

in which a is a quantity determined by observations of the heat capacity
of the solutions, and b and e are constants of integration. On the
assumption that a is independent of temperature, the e is connected with
the heat of dilution / by the formula

I ^ -- aS + e. (2)

If we use p» to designate the osmotic pressure at some definite tempera-
ture 6, of the solution, the above formula becomes

p p, 6 6 '- 69

- = - + a log- - e-TT' ' (3)

It is in this form, with certain slight modifications, that the formula
will hereafter be used.

2. Osmotic Pressure. — When the operation is considered by which a
small mass Am of solvent is forced out of a solution through a semi-
permeable membrane, thus reducing the volume of the solution by Av,
it is at once evident that the work done is not represented simply by
pAv. The solvent passes out against its own vapor pressure/, and if Au
represents the volume of the mass of solvent, negative work is done
represented by — fAu. Furthermore, work is done in compressing the
solution by raising the pressure on it to p, which is in excess of the

JJ^^i^'] RELATION OF OSMOTIC PRESSURE TO TEMPERATURE. 6$

negative work done when the pressure is removed, by an amount which
is proportional to A», and may be represented by hAv. The factor h is
equal to fip^/2, if we represent by /S the compressibility of the solution.
The work actually done in the osmotic pressure operation is therefore
represented by

(/>-/^+a)av = HAt;. (4)

The term containing h is generally insignificant, and we 'need to
consider it only when extremely accurate measurements are available.
In the examples which follow it amounts to only about i/ioth or 2/ioths
of one per cent, of the whole. The vapor pressure term is practically
independent of the concentration of the solution, and while it is insig-
nificant at low temperatures and with strong solutions, it may become
important at high temperatures with weak solutions.

Similarly the heat evolved and removed so that the osmotic pressure
operation may be conducted isothermally contains a term which expresses
the heat developed by compression. The total heat removed may be
represented by

(m + p)Av (5)

in which |a represents the heat developed while the solvent is forced
out, and v the heat developed by compression.

3. Boiling Point Cycle. — ^A reversible cyclic operation may be per-
formed by the aid of a semi-permeable membrane by (i) evaporating a
mass Am of solvent from a solution at the temperature 6»; (2) lowering
the temperature to ^ot the boiling point of the solvent; (3) liquefying the
vapor as solvent; (4) lowering the temperature to any temperature 6;
(5) admitting the solvent into the solution through the semi-permeable
membrane; (6) raising the temperature to 6,. When the equations of
energy and entropy are written out for this cycle, having regard to the
fact that the volume of the mass Am will be different at different tem-
peratures, it proves to be impossible to obtain a relation between the
osmotic pressures at the temperatures 6, and 6 which is symmetrical in
respect to those temperatures, as it should be, so long as the ordinary
and previously used definitions of a and / are accepted. If a and / are
referred to change of mass instead of to change of volume this difficulty
is removed. We shall adopt the definitions which yield the admissible
form of the relation. Accordingly we define a as the rate at which
the heat capacity of a system, consisting of solvent and of a solution
containing one gram-molecule of the solute, changes as the mass of the
solution increases by a transfer of mass from the solvent. Similarly we
shall define / as the rate at which heat is evolved in a solution containing

66 WILLIAM FRANCIS MAGIE. [^Sf

one gram-molecule of the solute as its mass increases by the addition
of more solvent. The experimental values of a and of / for aqueous
solutions will not be materially changed by this change in definition.

When these definitions are used it becomes convenient to replace the
term IIAv, expressing the work done by the osmotic pressure, by the

term PAm, in which

„ n Ai;

P =-T-.
p Au

The symbol p represents the density of the solvent.

Similarly we can set the heat evolved in the osmotic pressure operation
equal to JIf Am, in which

M + y Ar

M = r — .

p Au

By effecting the same change in a solution in two ways: (i) by the
immediate introduction of a mass Am of solvent, and the withdrawal of
the heat /Am evolved; (2) by the admission into the solution of the same
mass through a semi-permeable membrane, doing work PAm and supply-
ing heat JIf Am, it is easy to show that

Af = P - /. (6)

With these definitions, and on the assumption that a is independent
of the temperature, so that it may be considered constant in the integrals
that appear in the entropy equation, we obtain from the combination
of the energy and entropy equations, with the use of Exiuations (2) and
(6), the formulas

PP. B 6 - 6,

7 = T + ^ log- - e—TT-. (7)

P. _ _^_s - ff^fi

l*--^(p.~P.A«J- ^^^

6t difit 2 $g

In Equation (8), we have denoted by Xo the latent heat of the solvent at
the temperature ^oi by s and <r the specific heats of the liquid solvent
and of its vapor, by A^ the elevation of the boiling point of the solution,
or $9 — ^0; and by / the pressure of the vapor. The other symbols
have already been defined. The last term in (8) is insignificant in
practice.

4. Freezing Point Cycle. — By the use of a similar cycle carried through
the freezing points of solution and solvent. Equation 7 can be obtained,
and Equation 8 also, with the omission of the insignificant last term.
In Equation 8 the symbols represent those quantities, analogous to
those defined in connection with that equation, which are appropriate

Na*!^! RELATION OF OSMOTIC PRESSURE TO TEMPERATURE. 67

to the process of melting. A cycle of this kind has been studied by
Ewan.^ The approidmations which Ewan employs are such that his
formula will give results slightly different from those g^ven by Equation 7.
5. Test by Emden's Observations of Vapor Pressure. — ^These formulas
can be tested by applying them to Emden's observations on vapor
pressure.* Emden observed the vapor pressure over pure water and
over various aqueous solutions within a range of temperature from about
12° C. to 95** C. He found that the pressures over water could be well
represented by the formula proposed by Magnus

and that the pressures over the solutions were represented by a similar
formula, in which the constants b and c were unchanged, and the constant
/o took a new value/©'. Thus, according to Emden, the ratio

f u

at all temperatures, and von Babo's law holds for these pressures. Since
our formula for osmotic pressure shows that von Babo's law cannot
be accurately correct for all temperatures, we must calculate the vapor
pressures to which our formula will lead, in order to see whether or not
they also agree with Emden's observations within the limits of his
experimental errors. To do this (i) we first calculate from Emden's
formula the temperature t' at which the solution has the same vapor
pressure as water at certain definite temperatures t, and thus obtain a
set of values of ^' — ^ = A^, the elevations of the boiling point at these
temperatures. (2) We then select one of these values of A^ as correct,
preferably taking one for which the ratio /'// given by the experimental
numbers is the nearest to the ratio fo/fo used by Emden in calculating
the pressures over the solution for comparison with the observed pres-
sures. The particular ^ taken was in each of the cases following that
for which the temperature of the water was 80® C. (3) With this value
of AS we calculate PJO, for that temperature. In doing this the term
containing the vapor pressures may be neglected. The values of X©
employed in the examples following were calculated with a formula
kindly furnished me by Professor Harvey N. Davis, of Harvard Univer-
sity, according to which the latent heat of water vapor at any temperature
/° C. is given by

X - 92.98'[374.5 - ^r^".

(4) From this value of P,/tf, and the values of a and of / obtained for the

1 Ewan, Zeitflch. f. Phys. Chem., XXXI., p. 33, XS99.
> Emden, Wied. Ann., XXXI., p. 145, 1887.

68 WILLIAM FRANCIS MAGIE. ^SSS.

appropriate concentrations from observations of heat capacities* and
heats of dilution* the values of P/6 are calculated for the other tempera-
tures at which comparisons are to be made. The values of d, obtained
from Emden's formulas are sufficiently accurate to be used in the small
terms of the formula for P/$. If they are not known, that is, if we have
but one well determined value of ^ from which to calculate Pj6»i we
can calculate P/6 by successive approximations. (5) Having the values
of P/6 for different temperatures we now calculate back by the use of
the formula for Pj6t to the values of ^ for these temperatures. These
values generally differ only a little from those obtained from Emden's
formulas. Since for small temperature ranges the curve plotted with
the values thus obtained for Ad, measured off along the temperature
axis from the vapor pressure curve for water vapor, and the curve
similarly plotted with Emden's values of A^, will be closely parallel, the
true differences between the vapor pressure over the water and the vapor
pressure over the solution at the same temperature will be proportional
to that obtained from Emden's formula in the ratio of the two values
of AS. Thus we have

^^f_zi_ I - r/f

A6. /-// l-(/o7/o)*
using the subscript e to designate Emden's numbers. We may thus
calculate the true ratio /'// and from this, f. The vapor pressure thus
calculated for the different temperatures may then be compared with
the experimental values, or with those calculated from Emden's formula,
which agree with the experimental values within the limits of the experi-
mental error.

The following tables exhibit the results of such calculations. The
solutions for which we have at present the necessary information to
enable us to carry out the calculations are those for (I.) sodium chloride
5.067 parts, (II.) sodium chloride 10.096 parts, and (III.) potassium
chloride 10.051 parts, dissolved in 100 parts of water. The temperatures
of the boiling points of the pure water are given in Centigrade degrees.
In making the calculations, 273.1 was taken as the temperature of the
Centigrade zero on the absolute scale. The unit of energy is the calorie
and therefore the osmotic pressure is measured by a number which can
be reduced to absolute units by multiplication by /.

A glance at the figures in the last two columns of these tables will
show that the formula presented for the osmotic pressure furnishes
values of the vapor pressure over the solutions which agree very closely
with those calculated from Emden*s formula.

» Magie, this Review, XXV.. p. 171, 1907.
« Magie. thia Review, XXXV., p. 272, 191 2.

No. X. J

RELATION OF OSMOTIC PRESSURE TO TEMPERATURE.

I. ^0 from Emden* s formula.

Tamp, of Water.

0
20
40
60
80
100

Bmden's/ for Water.

4.5625

17.461

55.035

148.581

353.894

760.

I. ^.

II. 6B.

III. ^.

0.4549

0.9179

0.6100

0.5351

1.0800

0.7176

0.6219

1.2554

0.8340

0.7152

1.4439

0.9592

0.8151

1.6459

1.0932

0.9215

1.8609

1.2360

3. Values of a and e.

- 0.005671

- 1.806

II:

- 0.01662

- 5.405

III:

- 0.01376

- 4.3084

3. P/6, A6 calculated from formula for A^, /' calculated from this A^, /«' calculated from Emden*s
formula. The numbers in brackets are those on which the calculation was based.

Temp, of Water.

P/$.

A9.

/'.

0.00

35609

0.4470

4.418

4.414

36105

0.5313

16.897

16.89J

36292

0.6207

53.250

53.246

36235

0.7152

143.75

[35992]

[0.8151]

342,39

342.39

100

35601

0.9204

735.33

735.30

II.

0.00

70374

0.8852

4.279

4.269

72093

1.0631

16.354

16.337

72861

1.2490

51.511

51.492

72891

1.4424

139.03

139.02

[72474]

[1.6459]

331.11

100

71607

1.8552

711.2

711.1

III.

0.00

47929

0.6021

4.368

4.365

48949

0.7208

16.701

16.704

49230

0.8426

52.63

52.65

48947

0.9670

142.10

142.14

[48227]

[1.0932]

338.56

100

47163

1.2205

727.5

727.1

6. Comparison with Osmotic Pressures Obtained by other Methods: —
The true osmotic pressures p can be calculated from the formulas defining

70 WILLIAM FRANCIS MAGIE. [

P, given in § I. It did not seem necessary to present them. They do
not differ from the corresponding values of pP by more than two per
cent, in the extreme case of the highest temperature. They do not
show an exact proportionality with ^, as we should expect them to do,
if the osmotic pressure in all cases obeys the laws of gases. Numerically
they are so great, amounting, for example, to 40.28 atmospheres at o** C,
for the weaker sodium chloride solution, as to indicate the presence of
more than two molecules in the volume of the water used. This result
is inconsistent with the ordinary form of the dissociation hypothesis,
and points to some form of the association theory as necessary for its
explanation. Similar results have been found by Kahlenberg^ in some
cases by the study of freezing and boiling points.

Furthermore, the ratios of the true osmotic pressures of the two
sodium chloride solutions, instead of being equal to 1.9925, as they should
be if the osmotic pressure is proportional to the mass of dissolved solute,
or even less than that, if the more dilate solution is more dissociated,
and the osmotic pressure is proportional to the extent of dissociation,
are in all cases except at 0° C. slightly greater than 2, being generally
equal about to 2.010.

Kahlenberg's observations on the freezing points of sodium chloride
solutions may be used for comparison with the results calculated from
vapor pressure. The freezing points observed, plotted against the
quantities of salt dissolved, determine a straight line, from which we
find that the depression of the freezing point for our solution I. of sodium
chloride should be 2.990**. From this we can calculate the value of P«/d,
at that temperature. The value of PjB obtained for o** C. from this
freezing point is 0.0032086, and is about 10 per cent, lower than that
given in Table 3., I., for approximately the same temperature. If
it were the correct value it would require a vapor pressure over the
solution at o® C. of 4.431. The difference between this pressure and the
pressure 4.418 calculated by the formula for vapor pressure is within
the limits of experimental error; yet it cannot be said that Kahlenberg's
observation confirms our formula. DietericiV direct observation of
the vapor pressure over sodium chloride solutions at o** C. also gives
4.432 for a solution of the strength we are considering. The value of
the osmotic pressure calculated from Kahlenberg's freezing point is
in fair agreement with the dissociation hypothesis in this instance.
Kahlenberg's boiling points give also lower values in the case of sodium
chloride solutions than those calculated from Emden's observations by

» Kahlcnberg, Jour, of Phys. Chem., Vol. V.. p. 339. xpoi.
» Dieterid. Wied. Ann.. LXVII.. p. 859. 1899.

Na"i^'] RELATION OF OSMOTIC PRESSURE TO TEMPERATURE. J I

our formula. From Kohlenberg's series 3 we find the elevation of the
boiling points under 754 mm. pressure to be 0.820 for solution I., 1.720
for solution II. The boiling point elevations determined from the
formula of this paper for the same pressure are for solution I., 0.9194,
for solution II., 1.853. They are inconsistent with the dissociation
h3TX)thesis..

In the case of potassium chloride solutions Kahlenberg's Series 3
gives for the elevation of the boiling point of a solution like solution III.,
1.27. The value calculated from our formula is 1.2 14.

Palmer Physical Laboratory,
Princeton University.

72 THE AMERICAN PHYSICAL SOCIETY. [i

PROCEEDINGS

OF THE

American Physical Society.

Minutes of the Washington Meeting.

THE eighty-ninth meeting of the Physical Society was held at the National
Bureau of Standards, Washington, D. C, April 20 and 21. Two ses-
sions for the reading of papers were held on Friday and one on Saturday.

The following program was presented:

Thermal Expansion of Marble. L. W. Schad.

The Composition of Speech. I. B. Crandall.

Polarization at the Cathode in Oxygen. C. A. Skinner.

The Energy of Emission of Photo-electrons from Film-coated and Non-
homogeneous Surfaces: A Theoretical Study. A. E. Hennings.

The Electrical Conductivity of Sputtered Films. R. W. King.

Elastic Impact of Electrons with Helium Atoms. J. M. Benade.

The Loss of Energy of Wehnelt Cathodes by Electron Emission. W.
Wilson.

Theory of Ionization by Partially Elastic Collisions. K. T. Compton.

The Passage of Low-Speed Electrons through Mercury Vapor and the Ioniz-
ing Potential of Mercury Vapor. John T. Tate.

Amplification of the Photoelectric Current by the Audion. Jacob Kunz.

The Reflection Coefficient of Monochromatic X-Rays from Rock Salt and
Calcite. A. H. Compton.

The Measurement of *' A*' by Means of X-Rays. F. C. Blake and William

DUANE.

The Crystal Structure of Magnesium. A. W. Hull.

On the Ionization Potentials of Vapors and Gases. J. C. McLennan.

The Necessary Physical Assumptions Underlying a Proof of Planck's
Radiation Law. F. Russell v. Bichowsky.

High Vacuum Spectra from the Impact of Cathode Rays. Louis Thompson.

A New Theory concerning the Mathematical Structure of Band Series.
R. T. BiRGE.

The Wave-length of Light from the Spark which Excites Fluorescence in
Nitrogen. Charles F. Meyer.

A Measuring Engine for Reading Wave-lengths from Prismatic Spectro-
grams. L. G. HoxTON.

Natural and Magnetic Rotation at High Temperatures. Frederick
Bates and F. P. Phelps.

X^,^] THE AMERICAN PHYSICAL SOCIETY, 73

The Significance of Certain New Phenomena Recently Observed in Pre-
liminary Experiments on the Temperature Coefficient of Contact Potential.
A. E. Hennings.

Generalized Coordinates, Relativity and Gravitation. E. B. Wilson.

The Motion of an Aeroplane in Gusts. E. B. Wilson.

Optical Constants by Reflection Measurements. L. B. Tuckermann, Jr.,
AND A. Q. Tool.

The High-Frequency Absorption Bands of Some of the Elements. F. C.
Blake and William Duane.

A Proposed Method for the Photometry of Lights of Different Colors. III.
Irwin G. Priest.

The Use of a Thomson Galvanometer with a Photoelectric Cell. W. W.

COBLENTZ.

On the Occurrence of Harmonics in the Infra- Red Absorption Sf>ectra of
Gases. W. W. Coblentz.

On Friday afternoon President Millikan outlined to the Society the plans
of the National Research Council for enabling the United States government
to utilize the research ability of this and other scientific organizations for the
national defence. Considerable discussion followed.

At a short business session on Friday morning the following amendment
to the By-laws, presented from the Council, was adopted without opposing
vote. Change By-law No. i, section 2, to read as follows:

"For the election of a new member to the Society, either regular, honorary
or associate, or for the transfer of an associate member to regular membership,
it shall be necessary that a proposition in due form signed by two members
of the Society shall be presented at a meeting of the Council, and that at a
meeting of the Council occurring at least two weeks later, the person named in
such proposition shall receive the favorable ballots of a majority of the members
present. The council may, however, at its discretion make election to associate
membership at the same meeting at which the nomination is presented.**

The secretary made a brief report on the Pacific Coast meeting held at
Leland Stanford, Jr., University on April 8 under the direction of Pacific
Coast Secretary Lewis.

The attendance at the three sessions of the Washington meeting was about
no, 140 and 100 respectively. Visiting members were the guests of the
Washington members for lunch on both da>s. A subscription dinner at the
Cosmos Club on Friday evening was enjoyed by about fifty.

A cordial vote of thanks was extended to the Washington members for the
excellent arrangements for the meeting and the various courtesies extended.

The October meeting of the Society will be held at Rochester, N. Y., and an
excellent opportunity will be given to become acquainted with the important
industrial research laboratories there.

A. D. Cole,
Secretary.

74 THE AMERICAN PHYSICAL SOCIETY, ]MSSSi

Thermal Expansion of Marble.^

By Lloyd W. Schad.

A RECENT investigation on various American marbles shows that the
coefficient of thermal expansion of marble increases from about i X lo"*
at o® C. to 28 X 10"* at 300® C. After a marble has been expanded by heat
it does not come back to its original dimensions but a permanent increase
results, the magnitude of which depends upon the temperature to which the
specimen has been heated. A permanent increase in length amounting to as
much as 0.4 p>er cent, has resulted from heating a specimen to 300^ C.

National Bureau or Standards.
Washington, D. C.

The Composition of Speech.*

By I. B. Crandall.

THIS pap>er deals with the composition of average speech from sounds of
different frequencies, speech being considered as a continuous flow of
distributed energy, analogous to total radiation from an optical source. This
idea of speech is a convenient approximation, useful in the study of speech
reproduction by mechanical means.

Two properties characterize perfect 8p>eech reproduction: (i) The accurate
transfer of the language used, and (2) the preservation of the tone-quality of
the original speech. For expressing the ideal property of literal accuracy in
transfer various terms have been used, such as, "clearness," "intelligibility,"
"articulation,** and so on. The term "articulation** will be chosen to describe
this property of literal reproduction.

In reference to the other idea, namely, the preservation of the tone-quality
of speech, the term "naturalness** will be used. The idea of naturalness in-
cludes the preservation of the human or artistic quality of speech.

Consider first the relative importance of the different sf>eech frequencies
from the standpoint of articulation. Before we can determine this factor,
we must have a method of measuring articulation; this involves the choice
of a number of representative sounds, and the adoption of a testing routine
which will give the per cent, of such sounds accurately transferred by the
reproducing apparatus.* For a first attempt a list of the representative con-
sonant sounds in the English language has been taken, for experience has
shown that it is possible to identify most words in a given context without
taking notice of the vowels. The routine of articulation testing is rather
tedious and need not be gone into, as we are only interested in the result —

» Abstract of a paper presented at the Washington meeting of the Phjrsical Society, April
20-21, 1917.

• The method used is a development of that originally proposed by Dr. G. A. Campbell
in his article on "Telephonic Intelligibility." Phil. Mag.. 19, 1910, p. 152.

VojL. X.J j.^£ AMERICAN PHYSICAL SOCIETY, 75

the per cent, of consonant sounds accurately transferred by any given
apparatus.

It is possible to measure the relative importance of different speech fre-
quencies if we make a series of articulation tests using apparatus which com-
pletely suppresses certain frequencies, while at the same time the remaining
frequencies are perfectly reproduced. It suffices to state here that electro-
mechanical reproducing systems are available which have exactly these
characteristics.

Denoting the importance to articulation of any frequency p by the function
D(p)t we may consider the articulation x of a system which reproduces all
frequencies equally to be

Dip)dp = 1. 00.

By measuring x when different, limited ranges of frequencies are reproduced,
it is possible to find D{p), This method has been worked out in detail by the
writer and has yielded a good determination of the relative importance D{p)
of the different frequencies which compose the consonant sounds.

Coming now to the question of naturalness, the tone-quality of speech is
clearly defined if the relative amounts of energy associated with the different
frequencies are known: for this purpose we make use of another function of
frequency S(p) which indicates the energy distribution in speech. The com-
position of one unit of speech energy from energy of different frequencies may
be expressed by

S{p)dp - 1. 00.

Measurements of the relative intensities of different sounds are readily
carried out, and it is possible to determine the function S{p) in a number of
ways. One way would be to use apparatus similar to that used for the deter-
mination of D(p) in which certain frequencies were absolutely suppressed.
Instead of measuring articulation, we should measure the loss in loudness or
energy corresponding to a given suppression, from which data S(p) could be
easily found. Another way would be to experiment with systems which
reproduce all frequencies, but which overemphasize certain ranges of frequency.
Some rough experiments of this kind have been made from which preliminary
values of the function S{p) have been obtained.

The interesting thing, in the energy distribution in speech, is that the
vowels are the determining factors of this distribution, whereas the consonants
are the determining factors in the matter of importance to articulation. The
importance of the consonant frequencies in speech is thus utterly out of pro-
portion to the amount of energy associated with them.

On account of the fact that the energy in speech resides almost wholly in
the vowel sounds, it is possible to obtain the curve S(p) synthetically if the
energy distributions in the different vowel sounds are known. Making use of

76 THE AMERICAN PHYSICAL SOCIETY. [aSSS

Professor Miller's well-known results for energy distribution in the vowel
sounds and weighting each vowel for frequency of occurrence, I have con-
structed such a synthetic curve; the agreement between the synthetic curve
and the experimental values obtained from speech as a whole is practically
complete. More accurate data for the energy distribution will be offered in
a subsequent paper.

Because of the small amount of energy in the consonant sounds, they are
difficult to investigate; but experiments are in progress from which we hope
to obtain an interpretation of what has been called the importance of a given
consonant frequency. It is also hoped to give a complete treatment of the
reproduction of speech, based on the idea of the composition of speech given
in the present paper.

Research Laboratory of the American
Telephone and Telegraph Company

AND Western Electric Company, Inc.

Polarization at the Cathode in Oxygen*

By C. a. Skinner.

IN this pai>er were reported measurements of the polarization at the cathode
in oxygen, similar to those already published for hydrogen.'

These give the "transverse** current between two small electrodes placed
in the negative glow of a separate "ionizing** current, for given differences of
potential maintained between these "transverse** electrodes. The published
articles show that this applied difference of potential is practically concentrated
between the transverse cathode and the gas.

The results from oxygen are similar to those from hydrogen. Expressed in
terms of the apparent resistance at the transverse cathode, they are briefly
summarized as follows:

For a given ionizing current, the apparent cathode resistance remains
practically constant as the polarizing P. D. is increased from o to about 15
volts. Between about 15 volts and 140 volts the resistance is proportional
to the transverse current, giving the relation

where Vb is the polarization P. D. and 7 the transverse current density. Above
140 volts polarization P. D. the resistance rises to a maximum value then
slowly decreases — supposedly because of the increased production of ions in
the negative glow by the electrons escaping from the transverse cathode.

The deviation of the resistance curve from a straight line for polarization
potentials below 15 volts is ascribed to the temperature velocity of the ions
discharging to the cathode.

^ Abstract of a paper presented at the Washington meeting of the Physical Society, April
20-21, 1917.

* Phys. Rev., February and April. 19 17.

ItS^u] ^^^ AMERICAN PHYSICAL SOCIETY. 77

For a given polarization P. D. the apparent cathode resistance is practically
inversely proportional to the ionizing current density. From this law, the
calculated cathode resistance per cm.* at the transverse cathode is found to be
the same as that determined at the main cathode by wire sound measurements.

With transverse cathodes of fine wires, their apparent resistance per cm.'

is found to be, for the same polarization P. D., smaller than for plane cathodes.

Comparing diflferent sizes of wires, the cathode resistance is approximately

inversely proportional to the square root of their radius — if the same polarizing

P. D. be used.

University of Nebraska,
April, 191 7.

Elastic Impact of Electrons with Helium Atoms.*

By J. M. Bbnadb.

IT is well known that in ordinary gases an electron colliding with a molecule
loses all or nearly all its kinetic energy, even though its velocity is less
than the critical amount necessary for ionization. It has been generally
assumed and not without reason that in monatomic gases the impacts are
perfectly elastic. This means that when a collision occurs without ionization,
no vibration is set up within the atom and that no rotary motion is imparted
to the atom.

It is to be expected that the impinging electron will impart to the atom
a small velocity, thereby losing kinetic energy even in the case of perfectly
elastic impact. The influence of thermal motions of the atoms is negligible.
The average loss of energy assuming perfect elasticity of impact is 0.0054
volts when the velocity of the electron corresponds to a 20-volt drop, whereas
Frank and Hertz estimated an average loss of 0.3 volt per collision in helium.

An attempt has been made to determine the loss of energy per collision in
very pure helium with the result that the observed fraction of the electron's
kinetic energy which is lost is in close agreement with value calculated assuming
perfect elasticity. The agreement is such as to indicate that there is no
energy lost by inelasticity of impact.

The method employed was to determine the minimum voltage required to
cause ionization by collision in helium between two parallel electrodes at one
of which electrons were liberated by ultra-violet light. When the distance
between the electrodes, and the gas pressure, are small the necessary voltage
is the well-known minimum ionizing potential of helium (20 volts). But as
the distance and pressure are increased, the voltage, at which ionization
begins, rises indefinitely. At a pressure of 43.4 mm. and distance between
electrodes of 12.70 mm. the voltage necessary to cause ionization is no.

The fraction of energy lost by an electron at a collision is (i — 5) in the

> Abstract of a paper presented at the Washington meeting of the Physical Society. April
20-21. 1917.

78 THE AMERICAN PHYSICAL SOCIETY.

formula for the sum of a geometrical series

in which 5 is the energy of an average electron just able to ionize after making

n collisions, gaining A volts between successive impacts, n is obtained from

an extended table given by K. T. Compton's theory of ionization by collision

in the case of elastic impact. A is obtained by dividing the total voltage by n.

Since B — .99973f 5* can not be neglected unless n is very large, when (i — B)

becomes equal to A IS, In this case the gain between collisions is equal to the

loss at collision.

It has also been found that for helium XiPm (Stoletow's constant) is not a

constant quantity as in gases where collisions are inelastic. Values of XjPm

for helium range from lo to 27.

Palmsr Physical Laboratory,
Princbton Universfty,
April 7, 19 1 7.

The Energy of Emission of Photo-electrons from Film-coated and

NON-HOMOGENEOUS SURFACES: A THEORETICAL StUDY.*

By a. E. Hbnnings.

THIS study extends the theoretical considerations and examines analyt-
ically certain of the suggestions contained in a recent paper.' It deals
with the contact P. D. of a non-homogeneous surface brought into existence
by the formation of a photo-electrically active film upon a similarly active
homogeneous surface and the corresponding maximum energies of emission
of electrons from the components of such a composite surface when stimulated
by light of a given frequency. If the velocity of escape of electrons from one
surface or surface element to another is affected by the presence, in the near
proximity of one of them, of surfaces or surface elements of a different char*
acter, the logical conclusion seems to be that the potential to be applied in
order just to prevent the escape of electrons from one of the component portions
of a composite surface may not be at all the same as that required to prevent
their escape from a surface made up entirely of this component. An active,
initially homogeneous, surface upon which an active film electronegative to
it is forming might discharge electrons with abnormally high velocities, while
the electronegative film, at least in the earlier stages of its formation, might
discharge electrons with abnormally low velocities. Certain characteristic
features of the "distribution of velocity" curves for a series of metals or for
a given metal under difTerent surface conditions lend support to the point
of view which forms the basis of the analytical argument.

University of Saskatchewan,
April. 19x7.

^ Abstract of a i>aper presented at the Washington meeting of the Physical Society. April
20-21. 1917.

s Hennings and Kadesch. Phys. Rev.. N.S.. 8, p. 209, 19x6.

Vot.X.1
NO.Z. J

THE AMERICAN PHYSICAL SOCIETY.

The Loss of Energy of Wehnelt Cathodes by Electron Emission.^

By W. Wilson.

T was first shown by O. W. Richardson in 1903 that the thermionic current
from a hot cathode is given by the equation

where B is the absolute temperature and a and h are constants.

The constant h has the special significance that it is proportional to the
work done by an electron in leaving the surface of the body in question. This
work can be determined directly by measuring the difference in power required
to maintain a body at a certain temperature when it is emitting electrons
from when it is not. Richardson and Cooke and later Lester have obtained
values for tungsten which are in very good agreement with the values of h
obtained by Langmuir and K. K. Smith.

On the other hand, Wehnelt & Jentsch, Schneider, Wehnelt and Liebrich,
and Richarsdon & Cooke have all found that for lime-covered cathodes either
the effect is so small in comparison with other energy changes as to be com-
pletely masked or that there is no correspondence between the two quantities.

Since these experiments suggest that the mechanism of thermionic emission
from Wehnelt cathodes is different from that for pure metals further experi-
ments were made by the author to determine whether consistent results could
be obtained by using more stable cathodes.

If W is the work done by an electron in leaving the surface of a hot body
W •- hR^ where R is the gas .constant for one molecule.

If it is assumed that the work done by the electron is that done by moving
through a double layer of strength ^ we have W ™ t^ and <t> = hRle,

The method of Richardson and Cooke was used for the direct determination
of ^. The constant h was determined in the usual manner, the thermionic
current being measured with the cathode at different temperatures which were
obtained by means of an optical pyrometer of the Holborn and Kurlbaum type.

The following are the results obtained:

Pll.No.

t Defrtet.

bRI* Volts.

9 Volts.

27,200

2.34

2.39

30,100

2.59

2.54

23,500

2.02

L97

25,200

2.16

2.28

38,200

3.28

3.22

Filaments i and 2 were coated with BaO 50 per cent., SrO 2^ per cent.,

CaO 25 per cent.. Filaments 3 and 4 with BaO 50 per cent., SrO 50 per cent.,
and Filament 5 with CaO alone.

It appears that for Wehnelt cathodes the values ^ and hRje show a good

> Abstract of a paper presented at the Washington meeting of the Physical Society. April
ao-3i, 19x7.

8o THE AMERICAN PHYSICAL SOCIETY. [^S

correspondence. This is a strong point in favor of the view that the emission

of electrons from Wehnelt cathodes is due to a similar mechanism to that

causing the emission from heated pure metals. It is also a further proof of

the substantial correctness of Richardson's hypotheses to account for the

emission of electrons by hot bodies.

Rbsbarch Laboratory.

Western Electric Co., New York.

Theory of Ionization by Partially Elastic Collisions.*

By K. T. Compton.

IN a recent paper* the writer showed that, in a gas in which the electrons
lose no energy at impacts with molecules, except when these impacts result
in ionization, there should be a functional relation

P \P /

between the average number of ionizations a per electron per unit path, the
pressure p and the electric field intensity X, The equation expressing this
relation was derived from a simple assumption regarding the probability of
ionization at a collision.

Townsend has shown that, if there is any functional relation between a/p
and X/pi there should be a definite value for the Stoletow constant X/Pm for
the gas, where pm refers to the gas pressure at which the current through
the gas is maximum. Mr. Benade has shown that no such constant exists
in the case of helium, and also that the collisions in helium result in a small
loss of energy. His results further indicate that this loss of energy is entirely
accounted for by the velocity imparted to the molecules by the colliding
electrons, and therefore that the collisions are really perfectly elastic in their
nature.

The equations developed in this paper cover this case and also any case in
which the collisions are slightly inelastic. If v represents the total number of
collisions made by an electron while advancing unit distance in the direction
of X, and if c is the average energy, in equivalent volts, lost at a collision, then
AT' = A" — cv represents the net gain of energy per unit path. Thus this
case may be treated as a case of impact in which no energy is lost if we put

f-*(f)-*(^')-

If Vi = pN^ is the number of collisions per unit path in any direction,

ViVv* = (P + 5), where 6 is the ratio of c to the minimum ionizing potential Vq.

We can therefore put

* Abstract of a paper presented at the Washington meeting of the Physical Society, April
30-21. 1917.

lSo7^'] THE AMERICAN PHYSICAL SOCIETY. 8 1

+ 4N8

This equation is accurately verified by the experimental results of Gill and

Pidduck and it is found that c * 0.00268 volt, Vo = 20.99 volts, N = 8.7

collisions per centimeter path at a pressure of one millimeter.

Palmbr Physical Laboratory,
Princeton University.

The Passage of Low Speed Electrons through Mercury Vapor and

THE Ionizing Potential of Mercury Vapor.*

By John T. Tate.

THE present investigation is an attempt to determine with greater precision
the value of the ionizing potential of mercury vapor, and, in general,
to arrive at some explanation for the phenomena observed in connection with
the passage of low-speed electrons through mercury vapor. Franck and
Hertz have shown that the impacts between electrons and mercury molecules
become inelastic when the velocity of the electrons reaches a value corre-
sponding to a fall in potential of 4.9 volts. They interpreted their results as
indicating ionization of the mercury vapor at that potential but, as the writer
has pointed out, their results indicate nothing but inelasticity of impact. On
the other hand the results of Newman and of Gaucher would seem to indicate
the beginning of a weak ionization at that point. Some time ago, however,
the writer was able to show that if there were ionization at 4.9 volts it was
of relatively infinitesimal magnitude as compared with the ionization which
takes place at about 10 volts, and further showed that this was the minimum
potential required for the production of the many-lined spectrum of mercury.

The apparatus used in this investigation was similar to that used by Franck
and Hertz. The source of electrons was a lime-covered platinum wire which
formed the axis of two coaxial cylinders, the inner one of platinum net and
the outer of platinum foil. Varying accelerating potentials were applied
between the hot wire and net cylinder and measurements made simultaneously
of the total current between the hot wire and cylinders and of the current
reaching the outer cylinder against a retarding field. Measurements of this
type were carried out under varying conditions of vapor pressure of mercury
and of temperature of wire.

In general, the potential-total current curves show the characteristic dis-
continuity which takes place at a potential (usually greater than 10 volts)
depending upon the temperature of the wire and upon the vapor pressure.
The higher the temperature of the wire the lower the potential at which the
break occurs. Simultaneously with the sudden increase in current a visible

* Abstract of a i>aper presented at the Washington meeting of the Physical Society, April
30-21. Z917.

82 THE AMERICAN PHYSICAL SOCIETY. [

glow appears in the neighborhood of the hot wire and there is a copious forma-
tion of positive ions as indicated by the large positive currents between the
cylinders. With very hot wires the discontinuity could be made to take
place at potentials as low as 7.5 volts. After the ionization has set in it is
possible to decrease the potential considerably below the value at which the
ionization started before the current drops back, again discontinuously» to
its original low value. If we adopt the view toward which all the evidence of
the writer's experience points, that there is no ionization of the mercury vapor
of any appreciable magnitude until the electrons have an effective velocity
corresponding to 10.3 volts the explanation of the above phenomena is not
difficult. Owing to the inelasticity of impact at 4.9 volts, velocities much
larger than this value will be relatively infrequent if the vapor pressure is
high. As the applied potential increases, however, the probability of an
electron's acquiring a velocity of 10.3 volts is increased and more and more of
them will actually attain that velocity. As soon, however, as positive ions
are formed and are in a position to fall in to the hot wire before recombination,
they will, by their impact with the hot wire, cause the emission there of electrons
possessing initial velocities considerably higher than the initial velocities of
the purely thermal electrons. These high speed electrons will be all the more
able to ionize and hence, once started, the ionization will rush to completion.
The same reasoning allows us to explain the setting in of ionization, with very
hot wires, at applied potentials below the ionizing value. It is only necessary
to assume that under these conditions there are a sufficient number of high
speed electrons, due to local temperature variations, present to start the
ionization. Once started it will be self sustaining.

On the basis of the above theory it is to be expected that as the vapor pressure
is diminished it will become more and more possible for an electron to acquire
velocities in excess of 4.9 volts, and that finally a point might be reached
at which the increase in current due to ionization would take place at values
of the applied potential corresponding to the ionizing potential and that the
increase in current would be continuous, or nearly so. This was actually
found to be the case and was made the basis of an accurate determination of
he ionizing potential.

We might also expect, upon the above view, that at any given applied
potential, after ionization has set in, there would be two definite groups of
electrons emitted from the hot wire, one group consisting of electrons having
initial velocities corresponding to the thermal conditions of the wire, and the
other of electrons whose initial velocities are determined by the potential
through which the positive ions have fallen before striking the wire. The
initial velocities of this latter group would very probably correspond to
potentials differing from the applied potential, at least in cylindrical fields,
by a constant amount — the difference depending upon the amount of work
which must be done in removing an electron from the surface of the cathode.
The existence of these distinct groups of electrons is demonstrated by the

NaTxf '] ^^^ AMERICAN PHYSICAL SOCIETY, 83

presence of secondary maxima occurring at about 2 volts beyond each principal
maximum, beginning with the one at 9.8 volts, on the curves showing the
currents to the outside cylinder. No secondary maximum has ever been
observed after the first principal maximum at 4.9 volts. Further evidence
that these secondary maxima are due to electrons emitted by ionic bombard-
ment of the cathode is furnished by the fact that when conditions are such
that the curves for the total current are continuous these maxima disappear
or are relatively much less pronounced.

To determine the value of the critical potential at which ionization takes
place a series of current potential curves (see curve i of the figure) were taken
under conditions such that the increase in current was continuous. Under
these conditions the increase took place at a very definite potential (in the
curve shown, at 12.0 volts). This value must be corrected for initial velocities,
of course, and the accuracy of the present determination depends upon the
rather high precision with which initial velocity of the electrons could be
determined. Simultaneously with the above measurements of the total cur-
rent, measurements of the current to the outside cylinder against a retarding
potential of 3 volts were made. The curves showing the relation of this current
to the applied potential show the usual maxima which are very certainly known
to occur at effective potentials of 4.9 volts, 9.8 volts, etc. (see curve 2 of the
figure). If these two current curves are plotted on the same sheet we need
only observe the difference in potential between the second maximum of the
one, at 9.8 effective volts, and the sharp break in the other. In the particular
example shown this difference is just 0.3 volt giving as the value for the
ionizing potential 10. i volts. The average of five determinations is 10.3
volts and it is believed that this value is accurate to 0.2 volt.

Physical Laboratory,

Thb Univbrsiy of Minnesota.
April 4, 191 7.

The Kinetic Theory of Entropy.*
By w. p. Roop.

BOLTZMANN has demonstrated a parallelism between the entropy of
an ideal gas and the probability of its molecular state. What is it
whose probability increases when we increase the entropy of a gas by imparting
heat to it? There is no answer. For in the identification of entropy and
probability, an additive constant is neglected. This is found not to remain
constant when internal energy or volume suffer change.

Planck, by defining probability otherwise than in the usual mathematical
sense, eliminates the additive constant. For proof of the existence of a
relation between entropy and probability, however, he relies entirely on
Boltzmann. The relation established by Boltzmann is subject to the limita-
tion already mentioned.

^ Abstract of paper presented at the Stanford meeting of the American Physical Society.

84 THE AMERICAN PHYSICAL SOCIETY, ^SSS,

Entropy has a double significance: as a criterion of equilibrium, and as a
function of state. The entropy-probability relation appears to hold only in
connection with the first of these. Only through the establishment of a
general equation of state may kinetic theory be used in calculating entropy
as a function of state.

UNrVBRSITY OF CALIFORNIA.

On thb Ionization Potentials of Vapors and Gases.

By J. C. McLbnnan.*

FROM some experiments recently made in Columbia University by Pro-
fessor Bergen Davis and Mr. Goucher, a preliminary account of which
they have kindly sent the writer, it has been amply demonstrated that where
electrons having a velocity corresponding to a fall of potential of between lo
and 1 1 volts are allowed to bombard the vapor of mercury in a high vacuum
they are just able to produce in the vapor a definite and distinct type of
ionization.

This result constitutes a confirmation of the view presented by the writer*
in a series of communications, namely, that ionization potentials for the
atoms of mercury, zinc, cadmium, magnesium and possibly also for those of
other elements are given by the relation Ve = hy where 7 is the frequency
(1.5, 5). This view, it should be stated, was reached through experiments
made by my students and myself on (I) single line emission spectra, (II) the
absorption spectra of metallic vapors, (III) arcing potentials in metallic
vapors and (IV) from considerations arising from the theory of atomic structure
developed by Bohr.

If we accept the experiments of Bergen Davis and Mr. Goucher as conclusive
and as of general application, it follows that ionizing potentials may be calcu-
lated for those elements for which the spectral frequency 7 = (1.5, 5) is
known. These have been calculated on the basis of A = 6.585 X lo*"*' erg.
sec, and are given in Table II. In Table I. are collected the ionization
potentials for a number of elements which have been found by direct experi-
ment.

Table I.

Element. tonlsatlon Potential.

Helium* 20.5 volts.

Neon 16.0

Argon 12.0

Hydrogen 1 1.0

Oxygen 9.0 "

Nitrogen 7.9 "

Mercury vapor* 10.3 **

> Abstract of paper presented at the Washington meeting of the Physical Society, April

21, 1917.

* McLennan and Henderson, Proc. Roy. Soc., Vol. 91, p. 485, 1915* McLennan, Proe^
Roy. Soc., Vol. 91, p. 305, 1915. McLennan, Jour. Franklin Inst., p. 191, Feb. 19x6. Mc*
Lennan, Proc. Roy. Soc., Vol. 92, p. 574. 1916.

» Franck and Hert*. Ber. d. Deut. Phys. Ges.. Heft. 2. p. 44. I9i3«

* Goucher. Phys. Rbv., N.S.. Vol. VUL, No. 5, Nov., 1916. Bergen Davis and Goucher.
loc. cit.

«4
««

NoI"x^l ^^^ AMERICAN PHYSICAL SOCIETY. 85

Tablb II.

Blement.

A(y-i.5.5).

loniMitlon Potential Cal.,froin
y-(i.5,^).

Mercury

1187.98 A. U.
1319.95 " "
1378.69 " "
1621.7 '• "
2028.2 " "
2177.5 " "
2408.0 " "

10.45 volts

Zinc

9.40 "

Cadmium

9.004 "

Maflmesium

7.65 "

Calcium

6.121 "

Strontium

5.701 "

Barium

5.155 "

The values of those wave-lengths of the spectra for the different metals
whose frequencies are given by 7 = (1.5, S) excepting that for barium were
taken either from the inaugural dissertation by Dunz, Tubingen, 191 1, or
from that by Lorenser, Tubingen, 1913. The value X « 2,408 for barium was
deduced by the writer from the frequency 7 ■■ (1.5, 5)-(2, Pti which has been
estimated to be that of X « 8210.63. The experiments of Bergen Davis and
Mr. Goucher which have been referred to above go to show that the ionization
in mercury vapor which was observed by Franck and Hertz with electrons
possessing kinetic energy corresponding to 4.9 volts was not due to a direct
ionization of the atoms of the vapor, but had its origin in the metallic electrodes
of the ionization vessel and was due to the photo-electric action of the light
of wave-length X =■ 2,536.72 which was emitted by the mercury atoms under
the bombardment of the electrons.

Although the experiments have been arranged with much skill and appear
to be conclusive, another factor has recently been introduced by some experi-
ments of one of my students, Mr. R. C. Dearie. In these he has found that
with the vapors of mercury, zinc, and cadmium there is very marked absorp-
tion at the wave-length whose frequency is given by 7 « (2.5, 5)-(2, P),
It is not easy to interpret this absorption but it should be remembered that
in the spectra of the elements mentioned the wave-length which possesses
the greatest amount of energy is the one of frequency 7 =» (2.5, 5)-(2, P),

One inference on the Bohr theory which may be drawn from this result is
that the frequency 7 =» (2.5, S) with the relation Ve ^ hnf should give an
ionizing potential. Values calculated on this assumption are given in Table III.

Table III.

n^ment.

Mercury

Zinc

Cadmium . . . .
Magnesium . .

Calcium

Strontium^ . .
Barium*

> Estimated.

A(y-(«.5.^-(«,/^)

10,140.58 A. U.

11,055.4

10,395.17 "

11,828.8

13,038.0

13,717.82 "

13.251.00

II II
II
II II

A(y = «.5,5).

4,937.4
5,007.0
5.201.7
5,506.3
5,562.5
5,902.8
6,283.2

A(a.5,S)

2.51 volts

2.48 "

2.39

2.25

2.23

2.10

1.98

86 THE AMERICAN PHYSICAL SOCIETY. [

The value for mercury, it will be seen, is approximately 2.5 volts, which is
just one half the value obtained by Franck and Hertz.

At the New York meeting of the A. A. A. S. Professor Millikan described
some experiments made in his laboratory in which arcs were obtained in
mercury vapor with applied potentials of between 3 and 4 volts. Up to that
time arcs had not been obtained by the writer or his students in mercury
vapor, with less than an applied potential of between 10 and 1 1 volts, but the
point has now been re^amined by my students and myself, and we have
found that with mercury and cadmium vapors it is possible to strike arcs with
applied potentials less than those given by the relation V ^ h, (1.5, S)le if
tungsten filaments be used which are heated much higher than those used in
our first experiments.

Moreover with mercury vapor it has been found possible to sustain an arc
with applied potentials as low as 3.3 volts and with cadmium vapor with
applied potentials as low as between 2 and 3 volts.

It is difficult to interpret the fact that arcs can be maintained in vapors
with such low voltages, but it is possible that it may have something to do
with the results set forth in Table III. An initial emission of a radiation
stimulates by electronic bombardment combined with a photo-electric effect
due to the radiation reacting on the incandescent cathode might account for it.
To elucidate the matter, however, additional experimental work will have
to be carried out.

In Professor Bergen Davis' communication to me he pointed out that he
and Mr. Goucher had found that when mercury vapor was bombarded by
electrons, having a velocity corresponding to 6.7 volts it emitted light of
wave-length X = 1849.6. In this connection it may be of interest to recall
that in a paper by Mr. Thomson and myself,* recently published, it was
pointed out that when cadmium vapor was fed into a gently burning Bunsen
fiame the vapor was found to emit light of wave-length X = 3,260.17 but
that when the flame was made to burn strongly, light of wave-length
X * 2,288.79 was also emitted.

These two lines it will be noted are series lines analogous to the lines
^ = 2,536.72 and X ■= 1,849.6 in the mercury spectrum.

Mr. Ireton, one of my students, and I have also found recently that with
cadmium vapor heated in an exhausted quartz tube and bombarded by
electrons, whose speed is gradually increased, the line X = 3,260 comes out
with speeds of about 4 volts, but that when potential differences of about 6
volts are*applied the line X = 2,288.79 comes out as well on the photographic
plates. In this regard it will be seen that cadmium and zinc vapors act under
electronic bombardment in a manner analogous to mercury vapor, as Messrs.
Bergen Davis and Goucher have shown.

It was also found that with zinc vapor the line X = 3,076 came out when
the electrons attained a speed of about 4 volts and that when the electronic

* McLennan and Thomson. Proc. Roy. Soc., Vol. 92. p. 584. 1916.

Kol'u'] ^^^ AMERICAN PHYSICAL SOCIETY. 87

Speeds reached a value somewhere near 6 volts the line X = 2,139 also came
out strongly.

The Physical Laboratory.
UmvBRSiTY OF Toronto.
April 7. 19x6.

Optical Constants by Reflection Measurements.^

By L. B. Tuckbrman. Jr.. and A. Q. Tool.

THE properties of the light reflected from metals has been used as the
best available method of determining their optical constants. These
measurements, however, are vitiated by the effect of surface films, or transition
layers. L. Lorenz and Drude and, in a more complete form, Maclaurin,
have developed the theory of isotropic surface films, and Hebeker and Schulz
of doubly refracting surface films. Maclaurin has corrected Conroy's deter-
minations of the optical constants of steel from its reflection coeflicients using
his theory to eliminate the effect of the surface film, but his results are subject
to an error of several per cent., and of course allow of no independent check.

In trying to develop a practical method for eliminating the effect of surface
films from the measurement of optical constants by reflection it seemed best
to work first with transparent media where spectrometer measurements
afforded a check on the adequacy of the reflection method.

Using an old flint glass prism (v = 1.6537) whose surface showed a slight
deterioration and produced marked ellipticity, we were able to determine the
refractive index within 1/5 per cent, from reflection measurements, although
the value calculated by Brewster's formula using the principal angle of incidence
as the polarizing angle was 4}^ per cent, too small.

The measurements indicate that in the region from 40° to 70° incidence
the effect of the surface film is adequately represented by a modified Drude
theory — similar to Maclaurin*s theory, involving absorption in the surface
film, but neglecting second order effects. Below 40® and above 70® there
seems to be a systematic deviation which could be accounted for by second
order effects, if the accuracy of the determinations of the constants in that
region warranted it.

Drude*s original formula

Rp ^ cos (t? + x) r . -_J^__sin_^J^l

R. " cos (t? - x) L ''^^y/V+^tg^ - ^ J
was replaced by

Rp cos (t?

+ X) r .V ..V sint?/gt?l

jR« cos (t?

where (a) and (6) may be called the refractive and absorptive constants of

this film. For transparent films h should equal zero.

Univbrsity of Nbbraska.

* Abstract of a paper presented at the Washington meeting of the Physical Society, April
20-21. 19x7.

88 THE AMERICAN PHYSICAL SOCIETY. [

A New Theory Concerning the Mathematical Structure of Band

Series.^

By Raymond J. Birgb.

ACCORDING to Deslandres' law for band series, if the first frequency
difference of successive lines (Av) be plotted against integral values of
''m" there should be obtained a straight line either through the origin or very
close to it. The only serious modification of this theory, thus far proposed,
namely that by Thiele, has in the author's opinion been disproved definitely
by Uhler. In the case of the long {Ai) series running from the first head of
the 3883 CN band (the longest band series known) the author has found that
the curve in Av and m is accurately a hyperbola running through the origin,
with its real axis (on the particular scale of units used) making an angle of
13° + with the Av axis.

The final relation between frequency and m is most easily obtained by
actually taking the successive sums of the Av's. The agreement thus obtained
is Very satisfactory. For the 164 known lines of the Ai series the average
difference obs.-calc, with the "perturbations" included, is only .005 A.
Without these it is only one half as great. This is a greater accuracy than
has ever been obtained for any band series formula.

The hyperbola through the origin has four undetermined coefficients.
The final equation in frequency thus has five constants. The equation recom-
mended by Kilchling, in ascending even powers of m, if carried out to five
constants, is not nearly so satisfactory, but is unquestionably the best inter-
polation formula that can be used.

The corresponding series from the second and third heads give hyperbolae
of almost identical size and shape, but differently located. The integral of
the hyperbola, while not at all the proper functional form such that its finite
first differences {not its derivative) shall give a hyperbola, agrees to within
about .005 A. with the actual ZAv at all points in the series.

According to the above theory, a band series, if extended, will end when
the hyperbola again cuts the m axis, and would thus have a "tail" having the
same general appearance as the "head," i. e., a finite number of lines in a
finite space, whereas Thiele's "tail" required an infinity of lines. Thus weak
band heads may be the "tails" of strong bands running in the opposite direc-
tion. The author personally does not believe in the existence of such tails,
as all series continually decrease in intensity, after a certain point, those
studied ending in each case at about m « 168. If continued to the tails

they would have 220, 208, and 192 members, respectively, and would end at

• • •

about 3567.2 A., 3608.8 A., and 3647.5 A. There are no known tails at these
points.
It is not claimed, at this time, that the above formula will hold for all band

> Abstract of a paper presented at the Washington meeting of the Physical Society, April
20-21. 1917.

Xo'i^*] ^^^ AMERICAN PHYSICAL SOCIETY. 89

series. It seems probable, however, that it will be found to hold for all series
of the type of which those mentioned above are the best known and most
accurately measured examples.

DSPARTMSNT OF PHTSICS,

Syracuse UNivE&srry.

Generalized Co5rdinates, Relativity and Gravitation.*

By E. B. Wilson.

PROFESSOR WILSON discusses the relation between generalized or
curvilinear coordinates, the relativity of Newtonian mechanics, the
relativity of electrodynamics, and the new gravitation theories of Einstein.
The paper will appear in the Astrophysical Journal, 191 7.

The Motion of an AfiROPLANE in Gusts.*

By E. B. Wilson.

CONTINUING earlier work (Proceedings National Academy of Sciences,
Washington, 2, 1916, 294-297) Professor Wilson treats periodic longi-
tudinal gusts, gives a general discussion of resonance as applied to aeroplane
problems, takes up the limiting conditions arising in infinitely sharp gusts; then
turning to the lateral motion, determines the effect of single or periodic side
gusts and of yawing and rolling gusts. As the rolling gust is sometimes serious
because of its tendency to put the machine into a spiral dive, the motion of a
machine constrained to fly without rolling is treated and is shown to be
dynamically stable. The paper will be presented in detail to the American
Philosophical Society and offered for publication in their Proceedings.

Massachusetts Institute of Technology,
Boston. Mass.

The Significance of Certain New Phenomena Recently Observed in
Preliminary Experiments on the Temperature Coefficient

OF Contact Potential.*

By a. E. Hennings.

FROM the results obtained in these preliminary experiments it must be
concluded thjit in the case of none of the experimental work on the
temperature coefficient of contact potential hitherto recorded can it be posi-
tively asserted that what has been measured is the true coefficient. The
measured contact potential difference in vacuo between two copper plates,
the temperature of one of which was varied through a wide range, was found
to undergo abrupt as well as gradual changes. The nature and extent of these
changes depend on the degree of exhaustion of the containing vessel in such
a way as to indicate that gas films either mask the true effect or are themselves

the active agents which account for the phenomenon of contact potential.
UmvBRSiTY OF Saskatchewan.

> Abstract of a paper presented at the Washington meeting of the Physical Society. AprO
20-31, 19x7.

90 THE AMERICAN PHYSICAL SOCIETY. [|S»

Natural and Magnetic Rotation at High Temperatures.^

By Frederick Bates and F. P. Phelps.

PRECISION measurements have been secured for both the natural and
magnetic rotation of the plane of polarization in quartz for a tempera-
ture range from 20** C. to 1000** C. Quartz recrystallizes at about 575** C.
changing from a to /3 quartz. The curve for the natural rotation makes a
right angle turn at this point and shows a discontinuity. In contrast to the
natural rotation the magnetic rotation shows no change at 575** and has no
temperature coefficient. Similar measurements for the magnetic rotation of
iron films have also been made.

Bureau of Standards.
Washington. D. C.

A Measuring Engine for Reading Wave-lengths from Prismatic

Spectrograms.*

By L. G. Hoxton.

HE reduction of prismatic spectrograms is usually carried out by mean
of Hartmann's dispersion formula,

X = Xo +

5o — 5 '

especially for the routine work of astronomical observatories. Here X is the
wave-length, 5 the micrometer setting, while Xo, Sq and c are empirically deter-
mined constants, three wave-lengths at least being known.

The use of the formula, which involves computations with seven significant
figures, is laborious, even where a calculating machine is available, for the
evaluation of the last term involves the operation of division.

The present paper proposes a mechanical solution of this formula simultan-
eous with the procedure of setting on the lines of the spectrum plate. The
numerical results herein given must be regarded as preliminary because the
apparatus was crude for the degree of precision involved and hurriedly
assembled; but the accuracy actually attained is sufficient, in the author's
opinion, to show that no great difficulty will be encountered in constructing an
inexpensive attachment to any good measuring engine that would enable one
to read off wave-lengths with a precision considerably exceeding the precision
of setting upon the best of lines on a photographic plate.

The principle involved is that of geometrical projection. Use of this prin-
ciple for the graphical solution of the Hartmann formula has already been
proposed' and its mathematical exposition is, therefore, omitted here.

^ Abstract of a paper presented at the Washington meeting of the PhyBical Society, April
20-21. 1917.

*F. Henroteau, "On a Graphical Construction for Obtaining the Wave-lengths In Pris-
matic Spectra,** Monthly Notices Roy. Ast. Soc.. LXXVIL. i, p. 77.

UoT^] THE AMERICAN PHYSICAL SOCIETY. 9 1

The engine is an assembly of two interferometer beds (the mirrors being
removed) and a framework capable of rotating about an accurate pivot and
connected to one of the interferometer screws in such a way that settings
made upon the spectrum by means of this screw will move a hair-line fixed
to the framework into a position such that, when a setting is made upon this
hair-line by means of the other screw the latter screw will read off wave-
lengths. The adjustment of the engine for any given region of the spectrum
requires about a half-hour, and when once made, need not be repeated.

A trial was made in which the measures of 53 lines, previously made with
another engine and reduced by the Hartmann formula, were reproduced on
the setting screw of this engine, while the wave-lengths were read off as above
described. The results obtained by the two methods were then compared.
The differences, expressed in Angstrom units, are as follows classified according
to magnitude regardless of sign.

Differvnce-Range. Nvmb«r of

A. U. Initancaa.

From .000 to .010 inclusive 39

** .011 •• .020 ** 7

*• .021 •• .030 •• 5

" .031 " .040 " 1

" .041 •• .050 " ^

The plate concerned was taken with a 3-prism spectrograph of the Yerkes
Observatory and covered a range from X 4,434.168 to X 4,617.452.

Especial attention is called to the fact that this test involves the errors of

both screws, while for the direct measurement of a plate, the errors of but one

screw are involved. Further, the screws here employed were of a quality

inferior to that demanded by such exacting work. It is proposed to continue

this work with a fine screw in the near future.

University of Virginia,
March 8, 1917.

The Wave-length of Light from the Spark Which Excites

Fluorescence in Nitrogen.*

By Charles F. Meyer.

IT was shown by Professor Wood in 1910 that radiations from the spark were
capable of exciting fluorescence of air and other gases. Subsequent
investigations by Wood & Hemsalech, and Meyer & Wood, indicated that the
fluorescence was excited by light beyond the Schumann region. The wave-
length could not be determined, however, and some doubt still existed as to
whether the exciting light was really shorter than the limit reached by Schu-
mann. A method has been found of determining the wave-length of the

* Abstract of a paper presented at the Washington meeting of the Physical Society. Apri]
ao-ai, 19x7.

92 THE AMERICAN PHYSICAL SOCIETY. [^»

exciting light. A small grating is placed immediately below a fine slot in
one of the spark terminals, and the fluorescence in the path of the direct and
diffracted beams is photographed. From the angle between the two beams
the wave-length is calculated. Three determinations have been made with
two gratings which give a wave-length of in the neighborhood of 1,450 A. for
the light which excites the fluorescence of the water band 3,064 A. in nitrogen
containing a trace of water vapor. An interesting question arises concerning
the transparency of the air near a spark for this region of the spectrum.
Univbrsity of Michigan.

The Necessary Physical Assumptions Underlying a Proof of Planck's

Radiation Law.*

By Russbll v. Bichowsky.

IN order to prove Planck's radiation law by means of the quantum theory
only two physical assumptions need be made, first, that energy is absorbed
or radiated by a radiating system in quanta of Av, second that a radiating
system has the statistical properties of a perfect gas, ». e., that Maxwell's
distribution law holds for the distribution of the local values of the energy
among the codrdinates defining the state of the radiating system. (The usual
auxiliary assumptions such as Planck's oscillators, or Larmor's regions of
equal probability are not only unnecessary but misleading.)

Although these two assumptions are sufficient for deriving the Planck
radiation law both of them, and particularly the last, are very dubious, it being
almost unthinkable that a radiating system can have the statistical properties
of a perfect gas and yet not have the equipartition law hold. For these
and other reasons it seems necessary to give up at least the second of the
quantum hypotheses and to assume that the distribution of energy in a
radiating system does not obey Maxwell's law, that is, that in a radiating system
the distribution of the local values of the co5rdinates is a function not only
of the total energy of the system but also of some other variable. If we do
this and assume for definiteness that the distribution of the local values of
the generalized momenta is a function not only of the total energy E of the
system but also of the Helmholtz free energy A, and further assume that the
total energy of a radiating system cannot be less than a certain limiting value
£0 (£0 turns out to equal hv), we can, following the methods of Gibbs and
Ratnowsky, derive in a very simple manner the Planck radiation law and
moreover we can do this without assuming discreteness of radiant energy, with-
out contradicting classical mechanics (equipartition does not hold for systems
of this kind), without discarding infinitesimal analysis or without contradicting
thermodynamics or the direct experimental evidence of the photoelectric effect

that the hv law holds only as a limiting case.
Geophysical Laboratory,

Carnbgib Institution of Washington,
Washington, D. C.

1 Abstract of a paper presented at the Washington meeting of the Physical Society. April
20-ai, 1917.

Vol. X.J y^jg AMERICAN PHYSICAL SOCIETY. 93

The Measurement of "A" by Means of X-Rays.*

By F. C. Blake and Willlam Duane.

AT the New York meeting (February, 191 7) of the American Physical
Society, we presented a preliminary paper on the measurement of "A"
by means of X-rays. In this paper we stated that the values of "A" obtained
depended to some extent upon the interpretation placed upon the shapes of
the experimental curves. We have now succeeded in determining accurately
the corrections that must be applied to the measurements and have obtained
results that are quite consistent with one another.

The chief corrections appeared to be the following: (a) A correction for
the widths of the slits and of the source of the rays. The value of this correc-
tion we obtained by measuring the ionization currents in the instrument for
several positions of the micrometer screws that closed the slits.

(6) A correction for the depth of penetration of the X-rays into the reflect-
ing crystal. This correction we obtained by making the ionization chamber
slit very narrow, and, with the crystal in a fixed position, measuring the breadth
of the X-ray beam by moving the ionization chamber. If this is done on
both sides of the zero line of the instrument, we get values for both the loniza-
tion chamber angle and the crystal angle. The former should be twice the
latter, if what we may call the effective reflecting plane of the crystal coincides
with the axis of rotation. As a matter of fact, this never coincides with this
axis for all wave-lengths, the effective reflecting plane lying at a different
distance below the surface of the crystal for different wave-lengths. By taking
the difference between the measured values of the ionization chamber angle
and of twice the crystal table angle, we calculate the distance of the effective
reflecting plane from the axis of rotation, and from this we get a correction
that must be applied to the measurements of the wave-lengths.

In estimating the ionization chamber angle, we measure from the center
of the X-ray beam on one side to the center of the X-ray beam on the other.
If we take the distance between the two maxima, we get inconsistent results,
for the maxima do not, in general, coincide with the center of the beam on
account of the difference in the distribution of energy in the spectrum for
different wave-lengths.

We now have eleven measurements of the values of "A," determined by
using six slightly different methods of taking the data. These methods
may be briefly stated as follows:

I and 2. In these methods the ionization chamber slit is very wide and
there are two slits between the crystal and the X-ray tube, each of which is
narrow, thus permitting a very narrow beam of X-rays to strike the crystal.
In method i, the part of the curve representing the X-ray intensity as a func-
tion of the voltage, which is nearly straight, is extrapolated, and the point at
which the extrapolated curve cuts the axis of zero intensity is taken as the

* Abstract of a paper presented at the Washington meeting of the Physical Society, April
ao-2i, 1917.

i^ TME JLMERJCAN PHYSICAL SOCIETY, USS

r.-ur- i^ ▼-n^irt ^be X-raL\3 of wave-length corresponding to the center of the
3*r*:= "..jcisa* Nc sLi r:rTection is needed in this case. In method 2, ho'wc^^er,
T-i "L^atr rrt; n-ui: iT mhich the cin^-e actually vanishes, and make a cDerecdon
:t rre v^-:rs5- re lie sCit and source. These two methods gjA^c the same
^ 'W'-nrL JCss li-ai: one tenth of one per cent. In methods 3 and 4,
r -c- rcie tiirrw sat between the X-ray tube and the crx-staU and
■:2ie ircLJiiarr rtuniber slit very narrow also. The interpretations oC
l: ■•» n iiicdi>2s 5 arid 4 are similar to those in methods i and 2 respec-
-^ a anirrer zi iact- slit corrections in these methods do not turn out
ri:"i js^ -LTpt a* ir rseibods I and 2, but there is a certain disadvantage,
^ Z2X ZKsrzMziL zc the e5ecti\x re5ecdng p!ane in the cavstal most be
=iiiK-r wjzx rrssBDcrabue accuracy.
~*-rrr..-ca 7 iji£ t 5: ar«t diner from methods 3 and 4 except that we use
:ls; m— f-t ^-rreses:™^ the icniradon currents as functions of the wave-
tmcn AT Tinsrztjzz iiLiice- ir^^iead of the Cannes repcesenting the ionization
rs js A rinimra: :c the v.xtage at constant wa\'e4ei:gxh.
: Cwz:^ Zi.iut rrctiins the val-jcs ofi "k" obtained bv the above

• ■ w

■-i.'

-A.'

5 6357 X 10-"
$ 6348 X 10-"

6 tk-555 X 10-*
0 6354 X 10-"
^ tj354 X 10-"

McAi 6355 X 10""

^zs T^.r:~-:ri ireir ^i-je c< *i" cmaii>ec ir tbe abc\^ table does not
-— >*- -^^— ir:i.ni r-:=: -^>'^se ciicmlroi ;z c:"*^<r wa>^; foe issrancr, from the
ni=TT2 11^3 -r ^s*frzr- li t^ Ka.^k bv>i> $pe^rrr::=i. =sL::|: P-wick's radiation

■_ n- :r-.3^ ti^ rroic^^W:r:c tfiiNr:- '^si-c tbe Eisstein equation and

. - v_ — i ^-^rz=xnzil. rt^*.':^^; c^ t" =: F-." *r s :Vc-=-,::a lor the Rydberg
^-.— .:^, liin^ -i« diia »>b tailed rrors 5s>ec^-= aryu>"5is. The \'alues of
- -r ^i^^ e i_?c -e r-i^r Sfv:: v a. *. j^t^ni or :>e Sisas of t = 4-774 X lO~**
' :Z-^::l^ i -^1^ 1?^TC t^<^ >JLr-e vA:^-e o; t and r « = 1-770 X lo' we
-_ - ^-^.- — :^ 5v rr* <*c.--:^-~ • " ^' >J-l X :v^~*'> Ac^cvriing to Millikan's

, --— :^— -TT.: -^->:— TTrf r:?^ :**e vj. .-t^ o; ft ■ r>5^ X 10"^, and this seems to
_ -^ _^^ -^. 2' -A -^r-- r ^*:a AV^ »r r a- 'i^r *\,>crir3ents with X-rays
:i-.-» :z^-= r ' t ' z-^s ^\l .f> ^ 5: X :o " 1 .j..%? Ari Ku=t\ 6.53 X lO"*'

_ ^, ^-- J. ^^.. ^ ,•-•-. -' -ji.v-'s cv ,^ xji ,-^ v-< ' r/' we measured the

„ e -T- -' ~ ^"^ r*>r ^'^ •^'•i-j;>-> xV a* i^ iv r-.^rATx- voitmeter that we

_ .^ ^ .^ .-^- .; -i^-.»- -^ ;^ V, -v'' A * i^ resistance, as in the
7-^ ^-, -V "; >- - w xV *"-i^ X-rAv tube came from

Vw-^X.J j-ff^ AMERICAN PHYSICAL SOCIETY. 95

Any error in the estimation of the crystal spacing produces, of course, a
corresponding error in our value of "/r." We have taken for the grating
space (calcite crystal) the value 3.027 X lo"^ cm.
Harvard University.

The Reflection Coefficient of Monochromatic X-Rays from Rock

Salt and Calcite.^

By a. H. Compton.

ACCORDING to classical electrodynamics, the ratio of the energy of a
beam of X-rays reflected from a crystal to that of a beam incident
upon it is given by Darwin's formula (2) :

^^ Ei AS 2ti 4sin^cos^\wC»/ ^Y

In this expression Er is the energy in the beam of X-rays of wave-length
X which is reflected at a glancing angle d, while the crystal is rotated with
uniform angular velocity through an angle A^ which is large enough to include
all angles at which any appreciable amount of rays of this wave-length are
reflected. Ei is the total energy of wave-length X which falls on the crystal
during this time; N is the number of electrons per unit volume in the crystal.
fi is the absorption coefficient of the X-rays in the crystal; e is the charge
and m the mass of an electron and C is the velocity of light. The factor ^
depends upon the distribution of the electrons in the atoms of the reflecting
crystal, its value being approximately 0.76 for the first order reflection from
the cleavage planes of rock salt, and 0.75 for calcite.' The constant B depends
upon the thermal motion of the atoms of the crystal. It may be taken to be
2.6 in the case of rock salt reflecting molybdenum a rays, and 0.18 in the case
of calcite.

It is evident that by measuring the ratio of the reflected energy Er of wave-
length X to the incident energy Ei of the same wave-length, a test of this for-
mula may be made. In order to measure this ratio, monochromatic X-rays
were obtained by the reflection of a beam of X-rays from a crystal mounted
on a standard Bragg X-ray spectrometer. The source of X-rays was a
Coolidge tube with a molybdenum target, kindly supplied by Dr. Coolidge,
so that it was possible to obtain a comparatively intense beam of mono-
chromatic X-rays of wave-length 0.721 X lO"' cm. The monochromatic
beam thus obtained was reflected in turn by a crystal mounted on a second
spectrometer, and the intensity of the second reflection was determined by
the ionization method. This was compared with the intensity of the beam

> Abstract of a paper presented at the Washington meeting of the Physical Society, April
ao-ai, 1917.

■ C. G. Darwin, Phil. Mag., 27, 325 (1913) and 27, 675 (1914)-
» A, H. Compton, Phys. Rev., 9, 29 (1917).

96 THE AMERICAN PHYSICAL SOCIETY. lto«?

incident on the second crystal by removing the crystal and swinging around
the ionization chamber, so as to receive directly the monochromatic beam.

The quantity Er was measured by the total deflection of the electrometer
when the second crystal was turned with constant angular velocity through
an angle A^ past the angle of maximum reflection. The corresponding value
of Ei was the deflection produced by the monochromatic beam when it passed
into the ionization chamber for a time equal to that required to move the
crystal through the angle A^. The average value of R obtained in this manner
was 0.0050 dt 0.0003 deg.~* in the case of the reflection from a cleavage face
of calcite, and 0.023 =t 0.00 1 deg.~^ for a cleavage face of rock salt.

If in equation (i) iV is calculated assuming each atom to possess a number
of electrons equal to its atomic number, and /i is taken to be the usual absorp-
tion coeflicient (calcite 23.5: rock salt 18) we obtain R for rock salt » 0.040
deg.~* and for calcite 0.058 deg."*. It will be seen that for rock salt the experi-
mental value of this ratio is about one half the calculated value, and for calcite
is less than one tenth as large. The reason for this discrepancy is doubtless
due to the fact thai at the angle of maximum reflection a selective absorption
occurs, as has been predicted by Darwin^ from theoretical considerations,
and has been observed experimentally by W. H. Bragg* in the case of diamond.
The plausibility of this explanation is increased by the fact that if the reflecting
surface of a calcite crystal is roughened by grinding, the reflection coeflicient
is some three times as great as from a cleavage face. The grinding makes the
surface of the crystal imperfect, and thus greatly reduces the selective absorp-
tion.^ Experiments are in progress to make quantitative measurements of

the effective absorption coeflicient at the angle of maximum reflection.
University of Minnesota.

On the Occurrence of Harmonics in the Infra-Red Absorption

Spectra of Gases.^

By W. W. Coblbntz.

UNDER this title, Kemble* has given a theory to account for the occurrence
of absorption bands in harmonic series, in certain gases.
The observations, made by the writer, on HjS, SOi and NHs, being excep-
tions to his rule, Kemble concludes that the "extra lines may be due to im-
purities or they may be due to the nonlinearity of the law of force.'*

Following the procedure well known to spectroscopists (viz., if impurities
are suspected, to identify their absorption bands) it is found that these numer-
ous absorption bands cannot be accounted for on the basis of impurities. This
is very conspicuous in SOa which has very strong absorption bands. In the

» C. G. Darwin. Phil. Mag., 27, 675 (1914).
« W. H. Bragg, Phil. Mag., 27, 881 (1914).

* Abstract of a paper presented at the Washington meeting of the Physical Society, April
20-2I, 1917.

* Phys. Rev.. 8. p. 701, 1916.

So'if'J ^^^ AMERICAN PHYSICAL SOCIETY. gj

case of NHs, Baly* on the basis of a different theory computes sixteen absorp-
tion bands, between 3 and 14 /i» eleven of which coincide closely with observed
bands.

Water is one of the most unusual substances known, as regards its absorption
bands. The absorption spectrum* of water vapor consists of many fine lines
which, for water in the liquid state, coalesce into bands, the most conspicuous
of which occur at i, 1.5, 2, 3, 4.75 and 6/4. The apparent harmonic relation

01 the centers of gravity of the wide bands, comprising groups of these absorp-
tion lines is probably accidental. At lease, one would hardly consider them in
connection with the closely harmonic absorption bands of the simpler spectra
of other substances.

From a consideration of various phases of the problem of harmonics among
absorption bands the writer has come to the conclusion that the lack of agree-
ment is as much (if not more) the fault of the theory as it is of impurities in the
material examined.

Bureau of Standards.
Washington. D. C.
April 20. 191 7.

The Use of a Thomson Galvanometer with a Photoelectric Cell.'

By W. W. Coblentz.

THE development of a photoelectric cell by Kunz* giving a direct propor-
tionality of response with variation in intensity of the light stimulus
provides a simple instrument for investigations in the blue, violet and ultra-
violet parts of the spectrum, where the thermopile is operated with difficulty.

The object of the present communication is to call attention to the useful-
ness of a high resistance iron-clad Thomson galvanometer, instead of an
electrometer, in connection with the above-mentioned photoelectric cell.

Tests were made upon a two-coil instrument having a resistance of 1,300
ohms. Using a single swing of only 2 seconds and scale at 2 m. the current
sensitivity was i = 2.7 X io~" amp. A four-coil instrument, of 5,300 ohms,
and a heavy suspension, under similar conditions had a sensitivity of

2 = 6.2 X 10"" amp., or 8 X lo"*' amp. for a resistance of i ohm. From this

it is evident that a sensitivity of i X io~** amp. is easily attained, which is far

greater than would be required for transmission spectra investigations in the

blue violet and ultra-violet.

Bureau of Standards,
Washington,

April 20, 191 7.

» Astrophys. Jr., 42, p. 66, 191 5.

* Abstract of a paper presented at the Washington meeting of the Physical Society, April
20-21, 1917.

*Afltroph3r8. Jr., Mar., 1917.

ICX> THE AMERICAN PHYSICAL SOCIETY, ^SSS

It will be seen from the data of the last column that there is a progrersively
increasing diflference between our values and those of deBroglie, as we proceed
upward from element to element above bromine. This difference equals
approximately the error that one makes in measuring wave-lengths, if the
axis of rotation passes through the front surface of the crystal instead of
through the effective reflecting plane.

The curve representing the square root of the frequency as a function of the
atomic number plotted from our data is less inclined to the atomic numbers
axis than that plotted from deBroglie's data, but even our curve differs from
a straight line by slightly more than what we regard as the error of our
measurements.

Harvard University.

Second Series. August, 1917. Vol. X., No. 2

THE

PHYSICAL REVIEW,

IONIZATION AND EXCITATION OF RADIATION BY ELEC-
TRON IMPACT IN MERCURY VAPOR AND

HYDROGEN.

By Bbrgkn Davis and F. S. Gouchbr.

Mercury Vapor.

Introduction. — It has recently been pointed out by Van der Bijl^ that
the regular Lenard method for the direct determination of the ionizing
potentials of different gases and vapors, in particular mercury vapor, is
open to an objection which has not been considered by the experimenters
hitherto employing this method. The objection is based on the fact
that the positive charging up of the collecting electrode may as well be
due to a photo-electric emission of electrons from it, under the action of
the ultra-violet light emitted by the impacted atoms of the gas or vapor,
as to the formation of positive ions by impact. This fact could not be
determined from the curve shape since the number of radiating sources
(intensity of radiation) and the positive ions produced would both be
proportional to the number of impacts.

The possibility that this is the case in mercury vapor at a voltage less
than the true ionizing voltage is rendered highly probable, because of
the nature of the experimental results obtained by Tate,* Goucher,'
and McLennan and Henderson.^ Tate and Goucher have shown that
below the region of lo volts an effect is obtained setting in at 4.9 volts,
but that this is small compared to the effect occuring at 10+ volts.

It had previously been shown by Franck and Hertz* that impacts in
mercury vapor were elastic up to a certain minimum energy of the im-

» Phys. Rev., pp. 173-17S. Feb.. 1917.

* Phys. Rev., pp. 686-^87. June, 1916.
» Phys. Rev., pp. 561-573. Nov.. 1916.

* Proc. Roy. Soc., A, Vol. 91. 191 5.

* Verb. d. D. Pbys. Get., Vol. ix, p. 512.

t02 BERCEy DAVIS ASD F. 5. COUCHEML

parting electrons, \nz., 4.9 volts, and that at this voltage the electrons
lost their energ>'^ and at the same time emitted the radiation of wave-
length X « 2536.7 A., the frequenc>' of which was connected with the
\'oltage (4.9) by the quantum relation Ve « ir. They had assumed
that this loss of energy- was accompanied by ionization. McLennan and
Henderson extended this work and found that in mercury vapor this
single line X •* 2536.7 A. alone apparently was emitted up to a vahie of the
voltage slightly greater than 10 \'olts« but that the many lined spectrum
of mencur>* suddenly appeared if the \x>ltage was increased much beyond
this vahie« They also pointed out that thb \'ahie 10+ \x>lts is near that
calculated from the quantum relation, Ve ^ Af, when the frequency* taken
is that of the head or shortest wave-length of the Pascbem combtnatioa
series of the meixnirx* spectrum r « l^ — wP, viz,, 10-4 \t4cs; whereas
the line X « 2536.7 A, is the first or longest wave-fength of the second
subordinate group of this same series, \Hr„ f = 2^ — i*^ From
theoretical considerations then, in light of the Bohr theory, McLennan
was led to question whether iooiration reaKy took pLatce at 4.0 vcJts, or
or.ly at 10 -^ \x>Its at which the many Mned spectnim was esnirted. He
was led to conchide howex^er by his om-n ^IcLer.nan ai>d Ke%^ * cxperi-

merts cm, the <»Tvhjcti\"it\- of Aames in which mercury vapor was present

in a state of emirring or.V the Mrtt X « *536-7 A„ that ianizatkm really
did take place at this \xVItage and that therefore there seemed to be two
txpes of ionization in mercun- xapor. It shoiujc be noted hc»we\^er that
the flame conduv"*tv>n exp^eriments are open to the same criticism as those
emplcx-ing the direct method of Lenard.

It may be po:r.ted otut that there are no theorericaZ grounds for believ-
ing that there shooild be tmo tpes of ior.;7ariv-»n in mercury' vapor; nor
yet why, if a singk line was em i: red m-irhcuit ior*izarion at its corre-
sponding vo!ra^, the other lines m-erc no: emitted at their corresponding
voltages, instead of appearing a!:o$rerher when ionizarivTn had taken
p]aoe 35 the experiments of Mv"^lx*nnan seemed to indicate. McLennan
looked carefully for the other inten^^e line in the series, A-iz., \ = TS4Q A.,
which is the loneest wax-e-lencth o: the orincinal series, r=l.5; S^mP^
but wa< unable to find any trace of it.

Al! these facts rendered it hich!v desiraMc tr determine whether or not
the e fleets occuring K^low to. 4 vo-rs were due to ionizarion or to the
emission of ultra -violet lij^ht from the K^mharded atoms, and whether
or n'^t posit'AT ioni7ation actu«il!y to<->k piace at T0.4 volts. For the
purp.>?<:' of testing ^ith re*:ard to these possiM^itiesi. the following mexii-
tication of the ix*nard merho*.: ^^v^ nrorv^tsed hv one of us Goucherj.

* !*t«: Roy S»c.. A \'-^i o; p ?"«''i- loTt*,

No!"^] IONIZATION AND EXCITATION OF RADIATION. IO3

Apparatus. — ^The modification consisted essentially in the introduction
of a second gauze, C, Fig. i in the apparatus employed in the regular
Lenard method and described

in detail by Goucher; where Mnm|[ iimiM

A is the platinum equipoten-
dal surface electron source; B
the platinum gauze through
which the electrons from A
are accelerated ; D the collect-
ing electrode of aluminum.

The gauze C was of rather
large copper wire and coarse
mesh, and was supported by
the tight-fitting flange of brass
in the glass part as shown. ^^

Thearrangements were other- "^ —

wise quite the same as those
employed in the vessel used
by Goucher, with the excep-
tion of the palladium tube P

sealed in for the purpose of „ ,

admitting hydrt^n into the pi j

apparatus when desired.

All joints not of glass were ground and sealed with De Khotinsky
cement, the heater leads being also sealed tn with this cement.

The vessel was connected through a large | in. exhaust tube, a liquid
air trap, and a large stop-cock, furnished with a capillary by-path, to a
mercury diffusion pump of the Langmuir type, and to a McLeod gauge.

The electrical measurements were made by means of a suitably shielded
electrometer connected to D and sensitive to about 500 div. per volt.
The potentials were applied and maintained by means of dry cells and
suitable potentiometer connections, and were measured by a Siemens and
Halske standard voltmeter.

Method. — ^The procedure in making measurements was essentially that
employed in accordance with the Lenard method.

A field Vi was impressed between A and B, Fig. i, in such direction as
to accelerate the electrons from A through the gauze B\ a iield Vi wa6
maintained between B and C in such direction as to oppose the passage
of these electrons in the region BC and just enough larger than Vi to
prevent the electrons from reaching C. The departure from the Lenard
method consisted in the maintainance of a third and constant field Vt

I04 BERGEN DAVIS AND F, S, GOUCHER. [ISSS

between C and D throughout the measurements, but just as in the Lenard
method the rate of charging up of D was measured for different values of
the voltage Vi, Vt — Vi being maintained constant. The shape of the
current curve thus obtained and its intercept with the voltage axis was,
as in the case of the Lenard method, used as a basis for the interpretation
of the results.

The function of Vt was to control the field between C and D, it being
possible to maintain it either in the same direction as Vs or in the opposite
direction, thereby furnishing a means of distinguishing between a photo-
electric charging up of Z), and a charging up due to the production of
positive ions in the region BC. For if Vt be made smaller than Vt,
positive ions formed in the region BC will be able to reach D and conse-
quently will have a tendency to charge it positively whether Vt be either
in the same or opposite direction to Vj. Whereas, if the atoms of the
mercury vapor are stimulated to emit radiation, both C and D will be
in the path of such radiation, and would consequently be capable of
emitting photo-electric electrons, so that D would charge up due to this
cause, and the direction of this charging up of D would be determined
by the direction of the field Vt. If Vt were in the same direction as Vt
electrons would be extracted from D and driven to C (or through it into
the region BC beyond), while if Vt were in the opposite direction to Vt,
the emission of electrons from D would be prevented, and part of the
electrons emitted from C would be carried to D, causing it to charge up
negatively. The relative rates of charging up of D for these two direc-
tions of Vt, for any given voltage Vu would of course depend on the rel-
ative strength of electron emission of C and D under these conditions.
It is evident that they would follow the same law of increase with in-
creasing values of Vi, since the intensities of the radiation falling on C
and D should always be in the same ratio.

It is evident that if C and D are connected together we would have the
equivalent of the regular Lenard method, and we see why in this case
we would have no means of distinguishing, from the shape of the current
curves, as to whether it is caused by radiation from the impacted atoms
or to actual ionization of the gas. For, if the charging up of D were due
to radiation, the intensity of electron emission from CD would be pro-
portional to the number of atoms stimulated to emit radiation by impact,
whereas if it were due to positive ions from region BC, the number of
such ions would be proportional to the number of impacts resulting in
ionization.

The procedure then in the application of this method for the purpose
of distinguishing between these two causes, consists in maintaining a

JftT* 'J IONIZATION AND EXCITATION OF RADIATION. lOS

field Vt in a desired direction between C and D, The field Vt is small
compared to Vj. The field Vi and Vt are applied in their proper direc-
tions, the difference Vt — Vi being kept constant, and the rate of charging
up of D for different values of Vi is measured. The current voltage
curves, for Vt in the same or opposite direction to Vt, may be thus ob>
tatned and compared. Should the curves show a negative charging of
D when Vt is oppositely directed to Vt, we may conclude that at least
the effect of radiation is greater than that of ionization, and if the curve
continues to increase in the negative direction for increase of voltage Vi,
we must attribute this increase to impacts resulting in radiation, for the
tendency of ionization would be to cause an increase of current in the
positive direction.

The method can be further extended to the study of radiation alone,
by making Vt greater than Vt and tn the opposite direction to it, thus
preventing the positive ions that may be formed in the region BC from
reaching the collecting electrode D. The charging up of D in this case
will be due to the electrons emitted from C by the ultra-violet radiations
and carried to ZJ by the field Vi. The shape of the negative current curve
with different values of Vi will then serve as a basis for an interpretation
of the nature of the radiation emitted by
the atoms of mercury vapor when impacted
at various voltages.

Results. — For the purpose of making
measurements in mercury vapor liquid
mercury was introduced into the measur- .
ing vessel and contained in the part
marked "To B." Fig. i. The heat from ♦
the electron source A was sufficient to pro- ^
duce the desired pressure of mercury vapor g
for most measurements,*but the vessel was '
enclosed in a heat insulating box when
higher pressures were desired. The press-
ures usually employed were probably less
than .01 mm., estimating from the temper-
ature of the vessel at the time of making
the observations. ,kt«

The diffusion pump was kept running Fig. 2.

continuously to carry off any traces of

other gases than mercury vapor. The quantity of permanent gas pres-
ent was always too small to give a reading on the McLeod gauge.

Fig. 2 shows the current curves obtained in accordance with the regular

I06 BERGEN DAVIS AND F. S. COUCBER. [SSS?

Lenard method (C and D connected together), where (a), (b) and (c)
were obtained with decreasing electron emission from A, and over in-
creasingly wider range of voltages. Curve (a) shows the sharp break at
4.9 volts, and is the same kind of curve as that previously obtained by
Goucher. Curve (b) shows a marked increase of the current between 6
and 7 volts, while curve (c) shows this same increase and in addition a
discontinuity at 9.8 volts and a very sharp rise in current intensity at
10.3 volts. This value 10.3 volts is very close to 10.4 volts as cal-
culated from the head of the Paschen series. The discontinuity at 9.8
volts which occurs at twice the value 4.9, is what should be expected in
consideration of the elastic nature of the impacts of electrons having an
energy less than that due to 4.9 volts; and this would be true whether the
?nei^y lost at 4.9 volts were transferred into radiation or produced
ionization. This energy loss would occur again at twice this voltage,
viz., 9.8 volts, producing an increase either in the intensity of the radiation
or ionization at values beyond this.

The curves obtained, when the charging up
of D alone was measured, with a field Vt of
1.5 volts maintained between C and D, are
shown in Fig. 3. Curve (b) was obtained
with Vt in the same direction as Vt, and
curve (a) with Vt in the opposite direction
to Vj. Since curve (a) shows a negative
charging of Z>, increasing with increase of
applied voltage Vi up to a voltage of 10.3-I-,
^ where there is a sharp positive increase, we

can fairly attribute the effects below this point
I to a photo-electric emission of electrons from

I C. The production of positive ions in region

BC would cause an Increasing tendency to
make D charge positively with increasing
values of Vj. Since curve {b), up to I0.3-H
volts, shows a positive charging of D in ac-
pj 3 cordance with practically the same law of in-

crease as shown in (a), we can likewise attrib-
ute this part of the curve to photo-electric emission of electrons from
D. We may conclude from these results that the corresponding por-
tions of the curves obtained by the regular Lenard method (Fig. 2) were
also due to photo-electric action on the collecting electrode caused by
radiation from the impacted atoms of mercury vapor.
The strong positive charging of D in both cases, (b) and (a) Fig. 3,

KoTa.] IONIZATION AND EXCITATION OF RADIATION. lO/

above the value 10.3+ volts, we can attribute only to the production
of positive ions in the region BC, and since this value, within the limits
of experimental error, ts equal to the value calculated from the frequency
of the shortest wave-length of the spectral series, viz., lo^^ volts, it is
fair to conclude that this latter is the true ionizing potential of mercury
vapor.

Since the parts (a) and (b) of the curves (Fig. 3) are due to radiation,
the question arises as to the cause of the rise of these curves between 6
and 7 volts. Attention has been called to the fact that the other strong
line in the Paschen series is the wave-length X = 1849 A. This line was
sought for by McLennan but not found. The value of the voltage cor-
responding to this line as calculated from the quantum relation is 6.7
volts. It seems probable that the increase in the inten^ty of the radi-
ation between 6 and 7 volts b due to this cause.

Since the curves obtained show that the radiation occuring at 4.9
volts will produce a sharply defined discontinuity at twice this value
(9.8 volts), we would expect that if additional radiation were emitted
when electrons lose their enei^y at 6.7 volts, such electrons would be
capable of losing their energy a second time at twice this voltage, viz.,
13.4 volts, and consequently there should be a second rise in the radiation
curves beginning at this volt^e.

For the purpose of testing this point Vt was made large (about 20 volts)
and in the opposite direction to Vt, and a curve showing the negative
charging of D with increasing values of Vi over a range greater than 134
volts was obtained. The curves obtained with this arrangement of
voltages are shown in Fig. 4, where (o), (ft) and (c) are for decreasing elec-
tron emission from electron source ..4. The dotted lines show the points

io8

BERGEN DAVIS AND F. 5. COUCHER.

fSBOOND

at which rises in the curves should take place on the assumption that the
two lines X — 2536.7 A.andX « 1849 A. are produced at their respective
voltages. The shape of the curve (c) certainly indicates the existance
of such' effects.

The fact that there appears to be no marked increase in radiation at the
ionizing voltage (10.4) is quite significant. It indicates that the energy
of the impacting electron had gone into separating the electron from the
sphere of action of the atom, and in so doing had produced no radiation.
This suggests that the strong increase in radiation coincident with the
production of the many lined spectrum observed by McLennan and by
Richardson is due to recombination and not to ionization. It should be
noted that the pressure of mercury vapor in these experiments was small,
and that the chances for recombination are therefore small as compared
to the conditions employed by those experimenters.

The foregoing interpretation of the
results obtained for mercury vapor is
more easily understood by a consider-
ation of the schematic diagram, Fig.
5, where A represents the equipoten-
tial source of electrons and B the
gauze through which these electrons
are accelerated by the field Vi. The
arrows represent the directions in
which electrons would move in the
various fields. The extra gauze is
represented by C, and D is the col-
lecting electrode. The fields Vt and
Vt are maintained between BC and
CD respectively. The difference
Vt — Viis kept constant and just
large enough to prevent electrons from A reaching C. The field Vt is
also constant and arranged to carry electrons from C and D or vice versa
as represented by the arrows.

Consider the possible history of electrons with increase of voltage
Vi. At all voltages an electron may at this low pressure take a path as
represented by (i) in which it makes no collisions with the atoms. If it
does however make such collision and the energy is less than a given
minimum corresponding to a voltage Vo (4.9), the impacts are elastic as
shown by Franck and Hertz, and the path of an electron making such
impacts would be represented by (2), where the circles represent impacted
atoms. When Vi becomes equal to the minimum voltage Vo at which

Fig. 5.

No*^*] IONIZATION AND EXCITATION OF RADIATION. IO9

the electron will lose its energy, and if it is assumed that this energy will
appear as radiation, some of this radiation will fall on both C and D
causing photo-electric emission of electrons. These latter electrons will
move in the field V% according to its direction (a or i). This situation is
represented by (3) in the figure.

As Fi is increased beyond Vo an electron is capable of producing radi-
ating atoms in an increasingly wider range on either side of the gauze B,
thus proportionately increasing the intensity of the radiation reaching
C and D. This situation is represented by (4) and (5). When Y\
becomes equal to 2 Fo, impacts half way between A and B will cause the
electron to lose its energy, but it will be capable of again acquiring enough
energy to cause another atom to radiate in the region of the gauze j5, as
represented by (6) ; so that beyond 2 V^ some of the electrons would have
this double capacity for causing atoms to radiate and therefore we should
expect a corresponding increase in the intensity of radiation at a value
of Vi equal to2Vo.

If ionization takes place at or beyond B in the region BC, the positive
ions so formed would be carried to D irrespective of the direction of Ft,
so long as this latter is smaller than the fraction of Vt run through by the
positive ion before reaching C Therefore at this point we should expect
a tendency for D to charge positively irrespective of the direction of
Vu This situation is represented by (7). The transport of positive
ions to D can be prevented however by making V% larger than Vt so that
no positive ions can reach D from the regions BC as shown in (8). The
effects due to radiation alone can thus be studied even at large values
of Vi.

It may be objected that since Vt is slightly greater than Vu electrons
photo-electrically emitted from C or D would be capable of producing
positive ions in the region BC before those from A could do so. This is
of course true, but the number so doing is so small that it does not mask
the effect tmder these experimental conditions. This was shown experi-
mentally by increasing the difference Vt — Vi, and also by increasing the
pressure of the mercury vapor. There was an appreciable tendency to
charge positively below 10.3 volts. The point at which the curve started
to rise depended on (Ft — V'l), but in no case did it mask either the dis-
continuity found at 9.8 or 10.3-!- volts, which are of course independent
of (Fi — Vi). It may also be objected that since V% is large compared
with Vi and Fs, that it would be capable of producing ions and additional
radiation in the region CD. But since these ions and radiation could
only be produced by the electrons emitted from C by the action of the
radiation from the impacted atoms in AB and BC^ such ionization and

I lO BERGEir^AVIS AND F. S. COUCHER. [^»

radiation would tend to increase the magnitude of the breaks in the curves*
which occur at particular values of Vi.
The results of these experiments may be summarized as follows:

(a) Radiation is emitted without ionization at an impact voltage of
4.9 volts. This voltage corresponds to the frequency of the first line
X = 2536.7 A. of the Paschen combination v ^ 2p ~- mS, as has pre-
viously been poi'-^ed out.

(b) An increase in the intensity of the radiation takes place at an
impact voltage of about 6.7 volts. This voltage corresponds to the
frequency of the first line (X = 1849 A.) of the principal series v = i ,$S—mP
of this combination.

(c) Ionization by impact, without an apparent increase in radiation,
occurs at an impact voltage of about 10.4 volts. This voltage corre-
sponds to the head or shortest wave-length of this same principal series.

These results are of considerable interest when considered from the
point of view of the Bohr theory of the atom. The definiteness of the
results are due to the fact that the impacts in mercury vapor are perhaps
completely elastic. That is, an electron loses no energy at impact with
a mercury atom, unless either radiation or ionization is produced, in
which case the entire energy of the electron gofes into the radiation or the
ionization, and none is absorbed by the atom.

When the atom is impacted by an electron running through 4.9 volts,
its energy is transferred to an electron of the atom, lifting it we may sup-
pose from its equilibrium position to some ring farther from the nucleus,
and storing this energy in the potential form. Upon the return of this
electron to its equilibrium position this energy appears as radiation
(X = 2536.7 A.) in accordance to the relation Ve = hv.

When the atom is impacted by an electron having an energy corre-
sponding to 6.7 volts, an electron in the atom is lifted from its position
of equilibrium to some other ring still farther from the nucleus, and its
energy stored in the potential form. Upon the return of this electron,
its energy appears as radiation (X = 1849 A.) in accordance with the
above energy relation.

One considerable difficulty with this view of the process of emission
of radiation is that the other lines of this spectral series should appear
at their corresponding voltages. They apparently are not produced in
sufficient intensity to affect the curves obtained in these experiments.
The intensities of these other lines are weak compared to the intensities
of the two strong lines just referred to, when the radiation is observed
from the usual electrical discharge in mercury vapor. If the energy
emitted at each frequency corresponded to its voltage {Ve = Av), then

V&'I^'] IONIZATION AND EXCITATION i ff RADIATION. Ill

all the lines should be intense, and their intensities should progressively
increase toward the head or shortest wave-length of the series.

When the atom is impacted by an electron having energy corresponding
to 10.4 volts, an electron in the atom is lifted entirely from the atom
and removed from its sphere of influence. This electron is then free
and the atom is ionized. No radiation is then produced, as the electron
does not return to the atom. When the conditions are such that this
(or some other) electron may return to the atom (recombination) then
radiation is emitted. Experiments on the electrical discharge in mercury
vapor indicate that under these conditions not only is the Paschen spectral
series emitted, but the entire mercury spectrum including the visible.

Much of the phenomena of electrical discharge in gases indicate that
the greater part of the emission of radiation takes place at recombination
and not at ionization. We might mention for illustration the fact that
the most intense light from the usual vacuum tube discharge is emitted
from the cathode glow, where the electrical field is small» the concen-
tration of ions greatest and the recombination of the ions is far in excess
of their rate of production ; while on the other hand in those parts of the
discharge in which the ionization is in excess of the recombination the
emission of light is small. Some recent direct experiments of Child^
indicate also that a part at least of the emission of light from mercury
vapor is due to recombination of the ions.

Hydrogen.

Introduction. — It seemed desirable to apply the method employed for
mercury vapor to an investigation of hydrogen as well, on account of its
theoretical interest in connection with the Bohr theory of the atom.
The value of the ionizing potential for hydrogen has been found by
Franck and Hertz* and also Pavlow,' using the regular Lenard method,
to be II volts. This is in agreement with some recent work of Bishop,*
using the same method, who also has found by extending the current
curve over a wider ange of voltage, that there is apparently a second type
of ionization at 15.8 volts. Neither of these values are in accord with
the theoretical voltages, calculated from the Bohr Theory. The the-
oretical value yielded by this theory would be that corresponding to the
head or shortest wave-length of the series given by

where Tj =» i. Ti ■» i, 2, 3 which is the series observed by Lyman.

1 Phys. Rkv.. Jan., 1917.

* Franck and Hertz, Deutsch Phys. Ges., Vol. 15. 1913.

* Pavlow, Proc. Roy. Soc., Vol. 90, 1914.

* Bishop, not )ret published.

I 1 3 BERGEN DAVIS AND P. S. GOUCBER. [5SSi?

Using the value of N,

N - — ^— - 3.26 X 10".

as given by the Bohr theory, we can calculate the difFerent frequencies
of this series; and from the Ve = An relation can calculate the value of
voltage correspKjnding to the different members. This gives a value
10.2 volts as that corresponding to the first line or longest wave-length,
and 13.6 volts as that corresponding to the shortest wave-length. This
value then in the light of the Bohr theory should be the ionizing voltage.
It therefore seemed desirable to redetermine the values for hydrogen
by the application of this method to see whether or not the effects ob-
tained at 1 1 volts were due to radiation and not to ionization ; and whether
either the radiation or ionization had any connection with the above
values calculated from the Bohr theory.

The apparatus was the same as that used for the investigation of
mercury vapor. A stream of hydrt^ren was continuously passed through
the observation vessel by means of the palladium tube P. This was
heated by a gas flame and the pump was kept running. The pressure
of the hydrogen could be maintained as desired by regulating the flame
that heated the palladium tube.

The vessel was kept free from mercury vapor by means of liquid air
applied to the liquid air trap between the vessel and the pump.

The complete elimination of mer-
cury vapor could be tested by the dis-
appearance of the radiation effects
- characteristic of mercury vapor below

^ 10 volts which have just been de-

I scribed.

' Results. — With the pressure of

about .01 mm. and the potential Ki

small and arranged so as to draw

electrons from C to D, the potentials

Vi and Vt were arranged to give curves

of same type (o, Fig. 3) as found in

mercury vapor. With this arrange-

^^^ ment of Vi, Vt and Vt, if there were

p; g no ionization and only radiation, a

negative current would have been

observed. The curves actually obtained were all positive and are

shown in Fig. 6, where a, b, c and d represent results with diminishing

electron emission from equipotential source A. This indicates that

mS"^] ionization and excitation of radiation. 113

though there may have been radiation, the effects of this radiation were
more than overcome by the positive ions formed by impact in region CD.
These results confirm those of Franck and Hertz, who interpreted their
experiments as showing ionization by im-
pact at II volts. They also agree with
the results of Bishop who found a break in
the curve at about 15.8 volts, indicating a
second type of ionization at this voltage.
These curves (Fig. 6) might be due to ion-
ization alone or to a combined effect of ^
ionization and radiation, if the ionization I
effect were greater than the radiation effect.
The existence of a radiation without
disturbance due to ionization may be tested
for by an arrangement of potentials simi-
lar to that by which the curves (Fig. 4)
were obtained in mercury vapor. ' The po-
tential Vt was made larger than Vt (it was ^^^
made about 20 volts) and directed so as to F' 7
stop and turn back the positive ions com-
ing through C toward D. At the same time, the photo-electric electrons
emitted by radiation falling on C
would charge D negatively.

The curves thus obtained are
shown in Fig. 7, where a, b and c
are curves for diminishing electron
emission from the electron source
A . These reults reveal the striking
fact that there are two types of ra-
diation from hydr<^n. The one
type occurs at the ionization poten-

* tial of II volts and the other at
13.6 volts.

I The effects due to radiation may

I be increased and at the same time

those due to ionization may be de-

• creased by increasing the pressure
of the hydrogen. The curves in

Fig. 8. Fig. 8 were obtained with a press-

ure of .3 mm. Curve a was ob-
tained for Vi equal to 1.5 volts, and in the same direction as Vt, while

1 14 BERGEN DAVIS AND P. S. CQUCHER, [to»

a' was obtained with Vt at 1.5 volts and in the opposite direction to
Kj. The curve a shows the combined effects of ionization and radia-
tion, while curve a' shows the difference between the ionization and the
radiation effects.

The curves 6, b' were similarly obtained at lower pressures, where the
radiation effects were not so strong as at higher pressures. The ionization
and radiation effects nearly neutralize (6') until a voltage of about 15.8,
where the second type of ionization begins, when the ionization pre-
dominates.

At the pressure of .3 mm., the radiation and ionization effects just
neutralize from 11 to about 13 volts when the radiation predominates to
about 15.8 volts at which point the second type of ionization sets in, and
then the ionization effects predominate.

These experiments show the following facts:

(a) Both ionization by impact and emission of radiation occur at 11
volts.

(ft) A second type of ionization by impact without increase of radiation
occurs at about 15.8 volts.

These facts show a greater complexity than the simple Bohr theory of
the atom would predict, but are not inconsistent with it.

As has been indicated in a previous paragraph, by means of this theory
together with quantum relation the voltage corresponding to any fre-
quency may be readily calculated.

The voltage corresponding to the head or shortest wavfe-length of
Lyman series (Tj = i and Ti = «) is 13.6 volts. The voltage calculated
in the same way for the tail or longest wave-length of this same series
(ri = 2) is 10.2 volts.

There is thus a marked difference in the behavior of hydrogen and
mercury vapor. This latter gas showing radiation at a voltage corre-
sponding to the longest wave-length and ionization without radiation at
the head of the series. There is no radiation from hydrogen at 10.2
volts, which corresponds to the tail or longest wave-length of the series.
This may be due to the fact that the radiation of this frequency is very
weak or that some of the energy of the impacting electron is transformed
into kinetic energy of the hydrogen atom. This is quite probable since
the impacts in hydrogen are not elastic as in mercury vapor.

The occurrence of a new type of ionization by impact at 15.8 volts
instead of 13.6 volts which might be expected can also be accounted for
on the hypothesis that the hydrogen atom has a certain affinity for an

No!*a^l IONIZATION AND EXCITATION OP RADIATION. 1 15

electron. As the electron is displaced from the inner ring by the impact,
the radiation emitted on its return will correspond to the change in the
potential energy caused by the impact. When the impacting energy
is that due to 13.6 volts the electron is lifted to the outer ring or boundary
of atom and on its return emits the radiation of highest frequency. But
this electron when displaced to the outer ring will not be free as in case
of the mercury atom. If the hydrogen atom has an affinity for an electron
(non-elastic), it will require an additional energy to separate the electron
entirely from the atom (ionization). This additional energy will be
represented by the difference in voltage (15.8-13.6). This difference of
2.2 volts is thus a measure of affinity of a hydrogen atom for an electron.
An important result is the production of ionization at 11 volts. This
fact presents some difficulty in view of the Bohr theory, but it may be
due in some way to the diatomicity of hydrogen. We may perhaps
assume that at the 1 1 volts impact, the two atoms are separated one from
the other, and that the electron is taken away from one atom and attaches
itself to the other in this process, the one becoming a positive and the
other a negative ion.

It is hoped that we may be able to examine other diatomic non-elastic
gases, to determine if they behave in a similar manner.

Phcbnix Physical Laboratory,
Columbia University,
April, 191 7.

1 1 6 WILLIAM DUANE AND GERALD L. WENDT. [ISSS

A REACTIVE MODIFICATION OF HYDROGEN PRODUCED

BY ALPHA-RADIATION.

By William Duanb and Gerald L. Wendt.

TT is well known that the rays from radio-active substances produce
-■■ chemical reactions. They decompose water, whether the water is
in the liquid, solid or gaseous phase; they transform oxygen into ozone;
they split up hydrogen sulphide and ammonia into their elements;
they form hydrochloric and hydrobromic acid and ammonia from their
elements, etc.

The investigations described in the following pages were undertaken
in order to find out whether the rays produce an appreciable chemical
change in the purest hydrogen obtainable.

Sources of Radiation.

The most suitable source of radiation for the purpose appeared to
be a small "alpha-ray bulb" consisting of a glass sphere .5 mm. in
diameter, filled with radium emanation. If the walls of the sphere
are thin enough a large fraction of the alpha-ray energy passes through
them, producing an intense radiation in its immediate neighborhood.
Such a source also has the advantage that the radioactive substance
does not come into contact with the chemical reagents under investiga-
tion.

The bulbs used in our experiments were filled in the laboratories of
the Harvard Cancer Conunission by a method previously described.^
They usually contained about 35 millicuries of emanation.

Purification of the Hydrogen.

The hydrogen was prepared in the familiar zinc amalgam-platinum
cell with eight per cent, hydrochloric acid as the electrolyte. This
type of cell has been found to deliver very pure hydrogen in numerous
atomic weight determinations in the Harvard laboratories. Merck's
purest zinc bars were etched with hydrochloric acid, washed and im-
mersed in mercury, which had been washed in mercurous nitrate and
redistilled in air and in hydrogen. The amalgam formed the bottom

» Phys. Rev., 5, 311. 1915.

Vol. X.
No. a.

REACTIVE MODIFICATION OP HYDROGEN.

117

Fig. 1.

layer in a two-liter ground-glass stoppered Wolff bottle (i4, Fig. i)
electrical connection being made through a tube extending to the bottom
of the generator. The cathode was a sheet of carefully cleaned platinum
about 100 square centimeters In area
which had been coated with a deposit
of platinum black by the electrolysis
of a solution of chlor-platinic acid.
The hydrochloric acid was prepared by
distillation of a twenty per cent, solu-
tion of the purest commercial acid and
subsequent dilution of the purer frac-
tions of the distillate with distilled
water.

In order to prevent contamination of the hydrogen by organic ma-
terials, we avoided the use of stop-cock grease throughout. The stoppers
of the generator and of the purifying towers were surrounded by short
lengths of wide rubber tubing, which permitted the entire stopper to
be covered with mercury. This arrangement proved to be sufficiently
gas-tight. In place of stop-cocks, we substituted the U-tubes B and F
(Fig. i) filled with mercury, which could be opened by applying suction
at C and H. The hydrogen from the generator passed first through
three Emmerling towers filled with lumps of potassium hydroxide which
had been fused with a little potassium permangate to remove organic
matter. This freed the gas from add spray, chlorine, carbon dioxide
and a large part of its water vapor. Air, which had been dissolved in
the add solution, was next removed by passing the hydrogen through a
hard glass tube filled with clean asbestos fibres which had been soaked
in chlorplatinic acid solution and ignited to impregnate them with
platinum black. The tube was wound with nichrome ribbon covered
with asbestos and maintained at a red heat by an electric current. The
joints between the hard glass and the soft glass of the rest of the S3rstem
were ground to a close fit and surrounded by a jacket of glass filled with
mercury. This again gave a gas-tight joint without the use of grease.
(See D in Fig. i.) The gas was again dried in three towers of potassium
hydroxide lumps and passed finally through the U-tube £, about eighteen
inches in length and filled with phosphorus pentoxide, which removed
the remaining water vapor.

The hydrogen after this treatment could have been contaminated
only with nitrogen and the rare gases which might have been dissolved
in the add of the generator. To remove even these as far as possible,
we exhausted the whole system by means of a water aspirator to very

Il8 WILUAM DUANE AND GERALD L. WENDT. [toSS

dose to the vapor pressure of the acid solution. Hydrogen was then!
generated until the pressure reached atmospheric value. After three
repetitions of this exhaustion, the entire system was swept out by its
own hydrogen for forty hours before the conmiencement of the actual
experiment.

As a further precaution against some unknown impurity in this
hydrogen, we made a few experiments with hydrogen derived from the
electrolysis of a weak solution of potassium hydroxide containing a little
barium hydroxide to remove carbonate. The solution was in a cylindrical
vessel and a wide ring of heavy platinum wire formed the anode. Within
this hung a smaller glass cylinder tapering at the top and sealed to a
glass tube which in turn was sealed to the U-tube in place of the Wolff
bottle, A. A strip of platinum gauze sealed into the inner cylinder
formed the cathode. The cylinder extended ten inches below the level
of the cathode and anode in order to prevent contamination by the
oxygen liberated at the anode. This hydrogen was purified exactly
as in the first case, except that the system could not be exhausted.

We used still another variety of hydrogen, namely the electrolytic
hydrogen as supplied compressed in tanks by the International Oxygen
Company. A short piece of rubber tubing connected the tank to the
purifying system used before. The accumulation of water in the fourth
Emmerling tower in this case revealed some impurity of oxygen.

The Chemical Activity Imparted by the Rays.

In the first experiments undertaken we attempted to measure the
contraction in volume which should ensue, if Hs is converted into Hs.
A definite though small contraction appeared. On account of the size
of the effect a study of the chemical properties of the gas by some
dynamic method seemed more promising. In point of fact, the chemical
properties as revealed by the dynamic method are important in their
bearing on the observed volume change, so that the latter will be de-
scribed after the reactions.

In the first experiment a bulb of about 4 cm. diameter was coated on
the interior with a thin layer of sulphur by distilling flowers of sulphur
into it in a vacuum. The bulb was similar to the bulb B of Fig. 2, and
the bulb At containing the radium emanation, was similarly placed at
the center of B. Pure hydrogen entered through a side tube and after
passing slowly through the field of radiation passed out through a glass
tube, which dipped into a weak solution of lead acetate, and which held
in its mouth a strip of filter paper kept moistened by the solution. A
block of lead was so placed that only a minute quantity of gamma

JJ^af'J REACTIVE MODIFICATION OP HYDROGEN, II9

radiation from the emanation could reach the test paper or the acetate
solution. Wourtzel has reported the decomposition of hydrogen sulphide
by the rays, and inasmuch as ammonia, water and hydrobromic add
have been seen to be both decomposed and synthesized by the rays, it is
perhaps not surprising that the lead acetate paper showed a decided
blackening after the hydrogen had passed over it during one night.

This is the only reaction of a heterogeneous nature that has been
reported as produced by the rays. The question arises as to whether
the reaction is a consequence of the activation of the sulphur surface,
or of the hydrogen, or is an effect of the actual firing of the hydrogen
molecules into the sulphur by the bombardment of the alpha particles.
The last mechanism is not very probable because the rays penetrate
readily through hydrogen and comparatively few hydrogen molecules
are given high velocities. The question is capable of ready answer.
In the following experiment the bulb in which the hydrogen was radiated
did not contain sulphur, and the hydrogen after emerging from it passed
through a short tube containing redistilled flowers of sulphur and thence
over a lead acetate paper (see Fig. 2). A block of lead protected the
sulphur from the direct action of the rays. The paper was blackened
even more rapidly than in the preceding experiment. Repetition of
both these experiments with no change except the withdrawal of the
emanation produced no blackening. Without the sulphur there is also
no effect on the acetate paper.

Since the black coloration of the test paper can be due only to sulfide
the evidence is conclusive that the sulphur is being reduced by some more
or less stable compound, which is a stronger reducing agent than ordinary
hydrogen, and which becomes active after having been intensely bom-
barded by alpha rays. If this compound is an impurity in the hydrogen
it is difficult to imagine its nature. The reduction effect was obtained
equally well with the three quite different sources of hydrogen, A
number of tests were made in which the hydrogen, in addition to the
above described rigorous purification, passed through a glass spiral two
feet in length immersed in liquid air. Precisely the same results followed.
The only gases which could have survived this purification are helium,
neon, argon and nitrogen, and the acquirement of reducing properties
by any of these is harder to understand than the activation of hydrogen
itself. Finally whatever impurity may have been present was so very
dilute that the minute fraction of it that could have been acted upon by
the TBys would be unable to produce effects of the magnitude observed.
The evidence points unmistakably to an abnormal activity on the part
of the hydrogen itself.

I20

WILLIAM DUANB AND GERALD L. WENDT.

rSttOOND

This does not necessarily mean that a reactive molecule is being formed.
The hydrogen is subjected to a very intense ionization by the alpha rays,
and though the mobility of the gaseous hydrogen ion is high, large
numbers of ions undoubtedly are not yet recombined when they reach
the sulphur. It is entirely possible that the observed reactivity may
be due to these charged particles in the same manner that chemical
activity is the distinguishing mark of the hydrogen ion in solution.
To test this point we inserted a plug of glass wool between the ionization
bulb and the sulphur tube. Glass wool is exceedingly efficient in filtering
out ions, but in this case it had no effect on the result. This experiment
indicated the presence of a real modification of hydrogen, which is
chemically active. It cannot, however, be the same as Langmuir's
hydrogen because the latter not only cannot exist in the presence of
any considerable number of other hydrogen molecules but it is con-
densed completely by glass, especially in the form of wool.

To eliminate further the possibility that ions produce the observed
effects a tube was inserted between the ionization bulb and the sulphur
tube into which were sealed two platinum rods about four centimeters
in length and a centimeter apart on opposite sides of the tube (see F in
Fig. 2). These were connected respectively to the terminals of a battery

composed of two hundred very small
cadmium cells, and were therefore at a
difference of potential somewhat over
two hundred volts. The mobility of the
hydrogen ion in hydrogen is nine cms.
per sec. per volt per cm., and during the
time allowed for the hydrogen to pass
through this tube all the ions must have
been swept out of the stream of gas.
Nevertheless the paper was blackened precisely as before. Later some
tests were made using the large storage battery in the Jefferson Physical
Laboratory, but even the application of two thousand volts produced
no diminution in the chemical activity. The chemical activity, therefore,
cannot be due to the presence of ions.

We are dealing, then, with a more or less stable, reactive molecule.
Judging from the properties of Langmuir's hydrogen, which certainly
consists of free atoms, this new form cannot be monatomic. In Lang-
muir's experiments the free atoms as they were liberated from solution
in the tungsten or platinum filament deposited at once on the glass
wall of the evacuated bulb and formed there a thin layer. It was not
possible for him to obtain large quantities because the formation of

JJ^jf] REACTIVE MODIFICATION OF HYDROGEN. 121

more free atoms resulted only in their recombination with others on the
glass to form diatomic molecules. He did not obtain the atomic form
at all unless the pressure was so low that the atoms were able to pass
from the filament to the wall without encountering other molecules.
On the other hand, the modification we observed is fairly stable at
atmospheric pressure and it passes readily through a long plug of closely
packed glass wool — facts which exclude the possibility of its being
monatomic. The molecule is therefore polyatomic and larger than Hs.
It will be seen that all subsequent experimental data bears out this
interpretation.

An instance in point is the stability of this modification. Ozone reverts
spontaneously to oxygen at the rate of about 0.6 per cent, per minute,
and a polyatomic form of hydrogen would be expected to have a similar,
though perhaps somewhat greater, instability. We noted early in the
research that reducing the velocity of the stream of hydrogen diminished
the effect on the sulphur. If more than one minute elapsed between
the exposure of the hydrogen to the rays and its contact with the sulphur,
results were unsatisfactory. A few special experiments established the
fact that doubling this interval reduced by much more than one half the
chemical action.

The fact that this modification of hydrogen has a much higher boiling
point than that of ordinary hydrogen, being condensed even at the
temperature of liquid air, accords with the conception of a polyatomic
molecule. Fig. 2 illustrates the apparatus designed to investigate this
point. The hydrogen on emerging from the purifying system as pictured
in Fig. I passed through the glass spiral /, two feet in length, which was
kept completely immersed in liquid air throughout the experiment in
order to remove any condensible impurity. Passing into the bulb B,
the hydrogen was exposed to the alpha rays from the emanation contained
in the small bulb A. It passed then through the spiral H, through an
electrostatic field of four hundred volts per centimeter in F, through a
short plug of glass wool, then over the flowers of sulphur in the tube C,
and finally over the strip of filter paper, 2>, moistened with lead acetate
in £. Natural size photographs of the test papers resulting in this
experiment are shown in Fig. 3. Since it is not possible to reproduce
test papers from all the experiments carried out, those here presented
may be taken as typical. A coloration as feeble as that in (b) was
never accepted as positive evidence, (a) and (c) however are very
typical tests. The parabolic trace near the center of the strips corre-
sponds with the mouth of the glass tube where the hydrogen formed
bubbles and had a longer time to react with the lead salt. The test

122

WILLIAM DUANE AND GERALD L. WENDT.

Fig. 3a.

Fig. 3b.

Fig. 3c.

(a) was obtained by passing the hydrogen through the apparatus just
as indicated in Fig. 2, for six hours. At the end of that time another
paper was inserted and the Dewar bulb, /, filled with liquid air, was raised
to cover the spiral, H, through which the hydrogen passed after exposure

to the rays but before contact with the sulphur.
Nothing else was altered. In six hours the test
appeared as shown in (6). With the Dewar
bulb, /, lowered again a six hours' run produced
the test (c). Especial care was taken to maintain
the velocity of the hydrogen stream at twenty
cubic centimeters per minute throughout the
eighteen hours of the experiment. Although
there is a barely perceptible blackening of the
second paper, the active constituent of the
hydrogen has evidently been removed by the
low temperature. If this is a true condensation
the boiling points of Hs and of this hydrogen are
at least seventy degrees apart, and probably
more. It must be noted, however, that adsorp-
tion of gases is much increased at low tempera-
tures and that consequently the effect observed
may be due to an increased selective adsorption
of the active gas by the glass walls of the spiral,
though this is hardly probable in view of the
great excess of Ha always present. One other interpretation of the dis-
appearance of the activity is that the low temperature hastens the de-
composition of the larger molecule into Hs, but this is inconsistent with
the conceptions of kinetics.

In this test — as in every other — a blank run made without the
emanation produced no blackening. C? is a block of lead five millimeters
in thickness which prevented the beta and most of the gamma radiation
from reaching the sulphur or the test paper.

An attempt was made to obtain further information about the stability
of the active gas by interposing between the ionization bulb and the
sulphur tube a device for heating the gas to about 500® C. This con-
sisted of 40 cm. of fine platinum wire wound spirally on a quartz rod
supported by little quartz feet within a tube of soft glass sealed to the
rest of the apparatus. The terminals of the platinum wire were sealed
into this outer tube and connected through a rheostat with a iio-volt
circuit, and thus heated to dull redness. No information could be
obtained from this apparatus, however, because it was not found possible

Fig. 3.

No*^*] REACTIVE MODIFICATION OF HYDROGEN. 1 23

within the available period of one minute to Heat the hydrogen to the
temperature of the platinum wire and thereafter cool it again to a tem-
perature at which the ordinary hydrogen, unaided by the rays, no longer
attacked the sulphur. No blank tests could be obtained that did not
show some slight action of the ordinary heated hydrogen on the sulphur.
. A number of other reactions, besides the formation of hydrogen sul-
phide, were studied.

On substituting a tube of red phosphorus for the sulphur tube phosphine
was formed even more readily than the sulphide had been. A test paper
moistened with silver nitrate solution (and kept in th^ dark) blackened
considerably in the course of three hours. Here again a blank test
showed no action without the rays. It is interesting to note that
Langmuir's active hydrogen is completely destroyed when a little
phosphorus vapor is admitted into the evacuated bulb.

Powdered arsenic was also readily reduced to arsine, but this reaction
is slower than the two preceding, an exposure of about twelve hours
being necessary for a good test. The Gutzeit test was employed. The
filter paper strips were moistened with nearly saturated mei^curic chloride
solution. The coloration obtained was the usual brown shading into
yellow, both turning deep black when dipped into a solution of ammonia.

An interesting criterion of the reactivity of this hydrogen lies in the
possibility of its attacking metallic bismuth to form the hydride. The
formation of BiHs by this new hydrogen would have indicated a greater
reducing power than that possessed by nascent hydrogen. When pulver-
ized bismuth was placed in the contact tube a test lasting twenty-four
hours produced no mirror in a glass capillary tube kept at about 250^
by a luminous gas flame. No deposit of any kind was noted, so that the
bismuth does not appear to be reduced. This form, then, is not more
active than nascent hydrogen.

The new hydrogen attacks mercury.* This was discovered quite by
accident, when, in an experiment on the volume effect, a few droplets
of mercury were spattered against the interior wall of the ionization
bulb. After a day of radiation we noticed that the droplets were covered
with a crust of minute yellow crystals. Their form could not be deter^
mined with accuracy even under a magnification of 900 diameters.
Thereupon a wide glass tube which served as a sort of trough for about
six cubic centimeters of mercury was sealed in place of the contact tube
so that the hydrogen passed directly over the surface of the mercury.
In the course of two hours a faint yellowish scum appeared that on allow-
ing the hydrogen to pass over night, increased to a golden-yellow,

> As does Sir J. J. Thomson's Xa.

124

WILUAM DUANE AND GERALD L. WENDT.

wrinkled film which, as it became progressively thinner on the side re-
moved from the ionization bulb, showed two distinct series of interference
spectra. Not enough of the yellow substance could be obtained from
the deposit for an analysis. This is regrettable, because the result of
the test, unlike the others that have been mentioned, might have been
due to the presence of oxygen in the hydrogen and the formation of ozone*
In view of the preparation and purification of the hydrogen the presence
of oxygen is extremely improbable, and such as was present should, —
on the basis of Scheuer's work, — have combined rapidly with the hydro-
gen to form water. That so notable a quantity of oxide should have been
formed is hardly possible. The yellow substance was insoluble in water
and resisted attack by weak alkalis, but dissolved in HCl and HNOj.
On standing for a week it decomposed, leaving a dirty grey deposit which

on heating collected into droplets and distilled off, ♦. e.,
mercury. The experiment was repeated several times, but
m the course of two weeks all the yellow color regularly
disappeared. This is excellent evidence that the crystals
are a hydride of mercury, since such conduct would be ex-
pected of that substance while the oxide is very stable in
air. Two similar fractions of the yellow substance were
sealed in glass tubes, one in an atmosphere of oxygen, the
other in hydrogen, but no difference could be noted in their
rates of decomposition, except that both changed rather
more slowly than samples kept in the open air. When
gently warmed the yellow crystals broke up rapidly to give tiny drops of
mercury.

This hydrogen reduced neutral potassium permangate solution to
manganese dioxide. The absorption tube (Fig. 4) was used for this
test and for all other tests involving the reduction of substances in
solution. It is designed for the absorption in a minimal quantity of
reagent of a small quantity of a gas which is diluted with a large excess
of an inert gas. The gas enters through the tube at the left and bubbles
up through the reagent passing through the spiral also, in the form of
bubbles. It escapes at the top while the reagent which has been driven
up returns through the vertical tube. This gives very efficient absorp-
tion by small quantities of solution. The device, however, was not
necessary in the case of potassium permangate reduction because the
reaction is so rapid that the manganese dioxide deposited as a brown ring
around the bottom of the tube through which the gas entered the
absorber. Here, as above, a blsmk test without the emanation gave a
negative result.

Fig. 4.

j5S"^'] REACTIVE MODIFICATION OP HYDROGEN. 1 25

The same absorption tube was used in an attempt to reduce two
organic dyestuffs in water solution. Neither methyl violet nor indigo
carmine, however, were bleached by several days' passage of the hydro-
gen. The one, indigo carmine, involves the reduction of a ketone group
to a carbinol, the other, methyl violet, the reduction of a quinone
grouping. Both of these can be effected with nascent hydrogen, so that
this form of the gas would seem to be less active than the nascent.
The fact that the substances are in solution, however, may interfere
with the reaction. In a separate test the hydrogen bubbled through
water between the radiation and its contact with the sulphur. This
reduced the amount of sulphide only slightly, and water itself, therefore,
seems not to affect the active gas.

Mention may here be made of an experiment (to be described later)
in which a mixture of nitrogen and hydrogen was exposed to the rays.
Although a contraction in volume resulted, a test of the gases with
Nessler's reagent revealed only a small amount of ammonia. The
reaction between nitrogen and the active hydrogen thus seems to be
very slow.

A review of the facts thus far established shows that we are dealing
with a modification of hydrogen which is more active than the ordinary
form but less active than the nascent or Langmuir's monatomic states,
— ^properties which demand a polyatomic molecule larger than Hj. In
accord with this also are the physical instability and the boiling point.
In many respects the new gas is related to hydrogen as ozone is to
oxygen, but no method could be devised for determining its molecular
weight.

Change of Volume Experiments.

Fig. 5 represents the arrangement of the apparatus to detect the change
in volume due to the rays. Before the experiment the water-bath, C,
was not in place and the bulb A was connected by glass tubing with the
hydrogen generator and purifier while the bulb B was connected through
a small mercury trap with the open air. The mercury in the U-tube
was drawn into E by applying suction at G and the two bulbs thereby
connected. Pure hydrogen was swept through at a rate of about fifteen
cubic centimeters a minute for over twenty-four hours. The tube
leading from B dipped into a beaker of mercury through which the
hydrogen bubbled before reaching the open air. This was to prevent
the diffusion of air into the bulbs. When thoroughly swept out and
filled with pure hydrogen the bulbs were sealed off as shown in the
diagram. The water bath was then put in place and filled in order to
equalize the temperature in the two bulbs. This bath was constructed

126 WILLIAM DUANB AND GERALD L WENDT. [s^mi

of a shallow crystallizing dish through the bottom of which three holes

were drilled for the passage of the. three tubes. The dish was partly

filled with mercury and into this was set a tall cylindrical bottomless

beaker. Water was poured into this to completely cover the two bulbs.

Stirring was effected by a stream of air bubbles rising through the bath.

After standing at a fairly constant temperature for four or five hours

the mercury was allowed to rise in the U-tube, D-X,

by opening the stopcock F. His a trap to prevent

access of air into the U-tube from the bulb, £.

Since the bulbs B and A have nearly the same

volume the levels of the mercury in D and X are

nearly the same. Both positions are marked on

scales attached to the outside of the tubes. The

tubes D and X are capillary tubes with a bore of h^

Fig. 5, a millimeter, so that violent tapping is needed to

determine the true level of the mercury in each

arm. Once the level was thus determined the mercury could be raised

or lowered and accurately reset on the marks.

The emanation is then introduced into the small bulb, K. Fot this
purpose the emanation is compressed at the radium laboratory into a
fine capillary tube about two centimeters in length. This is inserted
into the bore of the stopcock, M, in such a way that it projects on both
sides. A sealed connection is then made at Jf with a Gaede rotary
mercury pump. The stopcock.s L and M are open and the bulbs N
and K are exhausted until no difference in level was exhibited by a Mac-
Leod gauge with a magnification of one thousand. The pump is then
sealed off from the apparatus shown in the diagram, leaving the bulbs
evacuated. The stO[>cock, M, is then turned and the fine tube containing
the emanation breaks, allowing the emanation to enter the bulb N,
The stopcock, M, is then sealed off at the constriction, P, and by opening
the cock, 5, the mercury rises in N driving the emanarion into K.
When the mercury has reached the neck of the bulb K the stopcock L
is closed. This is necessary to avoid changes in level of the mercury due
to the lai^e thermal expansion and contraction of the mercury in the
bulb N. With L closed these are applied to the air space in the trap, R.
In every case, the introduction of the emanation into K produced a
contraction in volume A, indicated by a rise of the mercury in the
capillary tube, X. The change in volume opposes and overbalances
the expansion due to the heating effect of the emanation.

The contraction In volume did not appear immediately. The pressure
change seldom amounted to a millimeter at the end of an hour.. At

l^^] REACTIVE MODIFICATION OF HYDROGEN. 12/

the end of three hours the pressure usually had fallen three millimeters,
and did not exceed this limit even after fifteen hours. Such a pressure
change involves a loss of 3/760 or 1/250 in volume, ♦. e., 04 per cent.
The bulb, A, contained about five cubic centimeters so that the actual
contraction amounted to 0.02 cubic centimeter.

The interpretation of the observed contraction is not quite clear, and
the phenomenon will be the object of further experimentation. The
change in volume appears to be rather larger than one would expect
from the number of ions produced by the alpha rays. On the other
hand, the effect may be due to some chemical reaction such as that
between the hydrogen and the mercury in the tube DX.

An interesting point now remains to be investigated, namely, whether
the active hydrogen is produced directly by the actual bombardment
of the atoms, or whether it may not be due to a breaking down of the
clusters of atoms which form around the charged ions.

In an attempt to decide this question, we performed the following
experiment. Hoping to be able to remove a large fraction of the ions
before the clusters could form around them, we ionized the hydrogen in a
very strong electrostatic field. Fig. 6 represents the ionizing chamber in
the apparatus. The purified hydrogen entered at D. CC represents a
cylinder of sheet platinum 15 cm. long and
4 cm. in diameter. The platinum rod B
lies along the axis of the cylinder and close
to the emanation bulb A^ A large storage Fig. 6.

battery was attached to B auid C, and pro-
duced a difference of potential of 2,000 volts between them. After
passing between B and C, the hydrogen flowed through E into the sulphur
tube.

The electric field had no appreciable effect on the result. The sulphide
paper appeared as black with as without the field.

If the field had destroyed or diminished the action perceptibly, it
would have been possible to draw some definite conclusions as to the
r61e played by the clusters. As no effect was observed, however, the
experiment must be regarded as negative and no conclusions can be
drawn from it, either one way or the other. Thus the mechanism of the
formation of the active molecules needs further investigation.

Conclusions.

It appears from the above described experiments that intense bombard-
ment by alpha rays produces in the purest hydrogen we could obtain a
certain capacity for entering into chemical reactions, which ordinary

128 WILLIAM DUANE AND GERALD L. WENDT. [l!S»

hydrogen at ordinary temperatures does not possess. For instance, the
radiated hydrogen combines directly with sulphur to form hydrogen
sulfide, with phosphorus to form phosphine, with arsenic to form
arsine, etc. It also reacts with mercury, and reduces neutral potassium
permanganate solution to manganese dioxide, etc.

This acquired chemical reactivity cannot be ascribed to the direct
action of the hydrogen ions, for the passage of the hydrogen through
glass wool and a strong electrostatic field (either of which would remove
the ions) does not destroy the acquired property of the gas.

The chemical activity may be due to the formation of H| by the alpha
rays. Certain characteristics of the active gas seem to point in this
direction. We have not obtained, however, conclusive proof of the
existence of Hs.

Experiments designed to discover whether the formation of the active
gas takes place during the actual bombardment by the alpha particles,
or subsequently, during the formation or disintegration of ionic clusters
turned out inconclusive.

The active modification can be removed from the hydrogen by passing
it through a tube immersed in liquid air.

A diminution of volume takes place in the hydrogen when it is bom-
barded by the rays. Experiments are in progress designed to discover
whether this is a primary action, or is due to some secondary chemical
reaction.

The active modification of hydrogen is not very stable, its life being
measured in minutes.

Harvard University.

No"^] COMPARATIVE STUDIES OF MAGNETIC PHENOMENA. 1 29

COMPARATIVE STUDIES OF MAGNETIC PHENOMENA. VIII.

A Study of the Joule and Wiedebiann Magnetostrictive Effects

IN THE Same Specimens of Nickel.*

By S. R. Williams.

THE photographic method, employed in former papers,* for recording
the changes in length produced in steel rods and tubes by a mag-
netic field (Joule effect), showed that small variations might easily be
overlooked by the ordinary telescope and scale method of observation,
because the latter process does not admit of a continuous record for all
field strengths from zero upwards. In the case of one steel rod a very
slight initial lengthening occurred and it was only by means of the con-
tinuous photographic record that it was detected. As will be shown in
this paper this mode of observing the changes in length in nickel rods
when subjected to a longitudinal magnetic field also enabled one to pick
out details which might otherwise have gone unnoticed.

Two nickel rods, numbered 5 and 6, each 80 cm. long and 0.397 cm.
and 0.318 cm. in diameter respectively, were employed in this investiga-
tion. Chemical analysis showed only very slight traces of iron present.
The change in length (Joule effect) and the twist (Wiedemann effect)
was observed in the manner described in the papers mentioned above.
Observations were taken on the rods in the condition in which they left
the manufacturer, because in this work it was a question of the com-
parison of magnetic phenomena on the same specimens. Heat treat-
ments and effects of stresses will be studied later. In the Wiedemann
effect a rod is magnetized circularly by passing a current through it
lengthwise while simultaneously a longitudinal field is applied. These
two fields superimposed upon each other give resultant lines of mag-
netization in the rod which are helical, as shown in Fig. i. In the Joule
effect changes in length occur along the lines of magnetization. This led
Maxwell* to regard the Wiedemann effect as a special case of the Joule
effect, because if changes in length do occur along the helices a twist
will result in the rod just as is found in the Wiedemann effect. One

* Preaented at the Cleveland Meeting of the Ph3r8ical Society, December 30, 191a.
« Phys. Rev., Vol. 34. p. 258, 1912; Amer. Jour. Sd., Vol. 36, p. 555, 1913.

* Maxwell, Elec and Mag.. Vol. a, p. 87, 2 ed., x88i.

I30

S. R. WILUAMS.

[

difference, however, seems to have been overlooked by Maxwell and
other observers, viz., that in the Wiedemann effect the direction of the
lines of magnetization is changing and in the Joule effect it is not. In

observing the Wiedemann effect it is customary to ap-
ply a constant circular field, C, Fig. i, first, and then a
longitudinal field, H, which increases from zero upwards.
If one of these components, C or H, Fig. i, varies then
the direction of the resultant field also varies. The larger
H becomes, C remaining constant, the more nearly 0, the
angle between H and the resultant field 5, becomes equal
to zero, and the direction of the resultant field, 5, more
nearly coincides with the impressed field H. This change
in direction of the resultant field in the Wiedemann effect
is, in my opinion, very closely associated with the fact that
the maximum twist in this effect occurs at lower values
of H than does the maximum elongation in the Joule
effect for the same specimens of iron and steel,* (See
also Fig. 2.)

From Fig. i it may be seen that 5, the length of the
spiral which forms the direction of the resultant field, is
equal to the length of the rod, L, multiplied by the secant
of the angle 0. For a spiral on the surface of the rod,

Fig. 1.

5 = L sec 0.

(I)

If change in length takes place along this spiral, then it is evident that
for small values of H the spiral will be very long, since 0 is almost 90®.
For this long spiral it would not take a large coefficient of magnetic ex-
tension with which to multiply S in order to obtain a comparatively
large twist and it would be very possible that by the time H had assumed
a value where the coefficient of magnetic extension was a maximum that
the value of S would have fallen off so that their product would be less
than at some previous value of H. This would mean a maximum twist
in the Wiedemann effect at smaller values of H than where the maximum
elongation in the Joule effect occurs. Due to hysteresis the lines of mag-
netization in the rod will not follow the lines of force applied to the rod,
but will lag* and therefore 0 in Fig. i for the lines of magnetization will
always be larger than 0 for the lines of the magnetizing force. A similar
line of reasoning will show that for nickel we should get a maximum of
twist in the Wiedemann effect although there is no corresponding maxi-

» Phys. Rbv.. Vol. 32, p. 281, 191 1.

* Ewing, Mag. Ind. in Iron, paragraph 140, and note, p. 238.

nS!"^] comparative studies op magnetic FHENOMENA. 1 3 1

mum elongation in the Joule effect. Apparently there is no way in which,
apriori, the real length of the spiral of magnetization may be accurately
obtained. Suppose, however, that there were no lag, and that we were
dealing with an extremely thin walled tube in Fig. i, instead of a solid
rod, or a thick walled tube, then we can get a relation between the Joule
and Wiedemann effects if the latter is a special case of the first as Maxwell
suggested. Multiplying S by J, the coefficient of magnetic change in
length, we will get the expression

SJ = JL sec if. (2)

which is the total amount of displacement the lower end of the spiral
takes in a direction along the spiral. The component of this displace-
ment along the length of the rod produces a change in length, d, Fig. i,
while the compooent, », normal to this, produces a twist in the tube. The
latter will be equal to

(JL sec 0) sin ^ ^ JL tan ^, (3)

where

tan * = CjH,

■^AJ^^e. (4)

the angle of twist which the lower end of the tube will experience due to
the Wiedemann effect, where r is the radius of the tube.

For certain specimens
of rather thick walled steel
tubes, this equation seems
to show qualitative rela-
tions very well. This is il-
lustrated in Fig. 2 for a
steel tube, B, used in pre*
vious investigations,' the
values of JL, curve L, B,
curve T, and OfJL, curve
A, are plotted with respect
to H. The curve for SjJL

is qualitatively what one

. . , . F'8- 2.

might expect from the re-
lation c ^ H tan 0, for rSjJL = cjH = tan 0. However when we carry
out the same curves for a steel tube, designated as C in the paper first
referred to, this «mple relation breaks down at moderately high fields,

> Phts. Rev.. Vol. 31. p. ti\, igii.

132 S. R. WILLIAMS. [

for experimentally we find 0 a negative value when J is still positive.
This must mean that other factors have entered to mask the simple
relation suggested by Maxwell. A change in the orientation of the ele-
mentary magnets with the change in the direction of the field might be a
possible factor. Here again is a very good illustration of the great com-
plexity with which we have to deal in trying to analyze magnetic phe-
nomena. Any complete analysis seems almost hopeless at times. The
publication of the results of investigation on magnetic phenomena should
contain very complete details of the modus operandi in securing the data,
for only as we know these minutiae of procedure may we make any
interpretations of the results that will mean anything. As a general
illustration, the magnetic induction of a substance like iron depends upon
many factors, such as temperature, stress, extraneous fields, etc., and
unless we know thoroughly the past history of that specimen, and how
it has been treated, it is hopeless to attempt to compare any results with
those obtained by another investigator who has used another specimen
which most probably has had still another past history. Only by a com-
parative study on the same specimens may we hope for progressive results.
As has been indicated, observations on the Wiedemann effect may be
made by applying a constant circular field first and then varying the
longitudinal field, or a constant longitudinal field may be applied to the
rod first and then the circular field varied from zero upwards. If we take
the value of the twist for the first case at definite magnitudes of the cir-
cular and longitudinal fields, and compare it with the twist for the same
magnitudes of circular and longitudinal fields, in the second case we find
they are not the same.^ This diflFerence is to be found largely, I think,
in this lag between the direction of magnetization and the magnetizing
force.

In one of the papers referred to, the fact was pointed out that in de-
magnetizing steel and nickel rods, particularly nickel, by means of a
decreasing alternating field sent through the magnetizing coil that the
specimens were always left in a magnetic condition such that the mag-
netization was always directed downward in the rods since they were
vertically suspended. This pointed to the probability of the elementary
magnets in the ferromagnetic substances settling down under the influ-
ence of the vertical component of the earth's magnetic field after they
had been shaken up by the demagnetizing process. We have an ana-
logous case in an iron rod which becomes magnetic by percussion when
placed parallel to the earth's magnetic field. Ordinarily pounding a
piece of iron demagnetizes it, especially if the rod is not in a magnetic

1 Honda and Nagaoka. Phil. Mag., pp. 6a, 63, Figs, ix and 13, Vol. 4. 190a.

Fig. 5.

S. R. WILLIAMS.

XS'af'l COMPARATIVE STUDIES OF MAGNETIC PEENOMENA, 1 33

field or is normal to it. In order to make sure of this point a series of
experiments was carried out in which various intensities of auxiliary fields
were applied to the specimens while they were being demagnetized.
The results were very conclusive. If an auxiliary field was applied to
the vertical rods in such a way that it was directed upward and its in-
tensity was greater than the vertical component of the earth's magnetic
field, then when the rod was demagnetized by a decreasing alternating
field in the presence of this auxiliary field the rod would altvays be found
magnetized upward instead of downward, as is the case when demag-
netized in the earth's field as the only auxiliary field. The presence of
these auxiliary fields plays a very important role in the magnetostrictive
effects and perhaps most of all in the Joule effect, being especially pro-
nounced in the case of nickel which appears to have its elementary mag-
nets very easily displaced if we may interpret magnetostrictive effects
as due to the behavior of the elementary magnets. Ewing and Cowan*
make note of the fact that ''it was only when the earth's magnetic field
was exactly balanced that this process gave complete demagnetization."
The presence of residual magnetism in the specimen of nickel was an
exceedingly sensitive method for determining the auxiliary field necessary
to balance the vertical component of the earth's field.

Inasmuch as the changes in length due to a magnetic field could be
registered photographically, a further study of these auxiliary fields
was made by investigating what kind of changes in length occurred when
auxiliary fields other than the earth's field were present during the process
of magnetizing the rods. In Figs. 3-6 are shown various photographs
taken under different conditions of auxiliary fields. In all cases the
graphs may be thought of as moving from the readers right to left behind
the lower end of the suspended rod which traces its motion on the film
as the field H varies in value. When the trace moves toward the top
of the graph there is a shortening of the rod and when the trace is directed
downward the rod is elongating.

In At B, C and 2?, Fig. 3, is shown the effect of the earth's magnetic
field as an auxiliary field upon the nickel rod No. 5. A is for a field in
the solenoid which is directed upward and opposed to the vertical com-
ponent of the earth's field. It will be noticed that there is an initial
lengthening followed by a contraction just as is found in steel. Graph
B shows the change in length when the field in the magnetizing coil is
directed downward, ♦. e., in the same direction as the vertical component
of the earth's field. In this case there is no initial lengthening, the rod
contracts from the start. C and D are repetitions of A and B respectively

1 Ewing and Cowan, Phil. Trans. Roy. Soc. London, p. 326, Vol. 179. 1888.

134 ^' ^' WILLIAMS. [;

and show what happens however often the magnetizations are repeated.
Before each graph A^ B, C and D, the rods were demagnetized by a de-
creasing A.C. field, 60 cycles per second. In each case, however, when
the magnetizing force of the solenoid was opposed to the vertical com-
ponent of the earth's field, there an initial lengthening occurred for nickel,
as we find it in steel also.

In Fig. 4, graphs A, B, C, D, E and F, have been taken without de-
magnetization previous to each record, and each graph was repeated with
the magnetizing force of the solenoid in the same direction. A, B and
C are with the field up while D, E and F are with the field down. A
was preceded by a field which was in the same direction as A, and D
also succeeded a field in the same direction as D. It is to be kept in
mind here that the auxiliary fields present are the earth's field and the
remanent field of the rod, the latter evidently being the predominant
one as shown by a comparison of il, -B, C, D, E and F of Fig. 4 with At
B, C and D of Fig. 3. In G and H, Fig. 4, is shown the effect of reversing
the field, first up and then down, without demagnetization before each
graph. This again shows the predominance of the remanent field of the
rod. Graph G was preceded by a field in the opposite direction, ♦. e.,
a field similar to H,

In order to produce other auxiliary fields a single layer of insulated
wire was wound over the whole length of the magnetizing coil. By means
of this auxiliary coil magnetic fields of various intensities and directions
could be imposed on the rod while the magnetizing force of the main
solenoid was being varied. In graphs /, /, K and L, Fig. 4, is shown the
effect of an auxiliary field produced by the additional coil and of such a
magnitude that it was about three times that of the vertical component
of the earth's field, 2.18 gauss. In graphs / and / the auxiliary field is
directed upwards while in K and L this field is reversed. / and K are
for fields in the main coil which are directed upwards while / and L are
downward. Here again it will be seen that whenever the auxiliary field
is opposed to that of the magnetizing field of the main coil, that then we
get an initial lengthening in nickel. Demagnetization occurred before
each graph here also. Finally in graphs £, F, G and H, Fig. 3, are shown
the changes in length when the vertical component of the earth's field
is just annulled. In this case there is no initial lengthening but a decrease
in length for all field strengths. Demagnetization occurred before each
graph in this set.

In like manner the nickel rod No. 6 was studied and found to be very
similar. In fact, rods 5 and 6 came the nearest to being alike magnetically
of any two specimens I have ever found. This study of these two rods

NaTa*!^*] COMPARATIVE STUDIES OF MAGNETIC PHENOMENA. 1 35

has been concerned only with small field strengths, because the mag-
netic behavior for these fields is of more interest from the standpoint of
what is happening inside of the specimens when they are magnetized,
than for higher fields. It will be noticed that in all of the graphs for this
paper there are six vertical lines to each graph. These are to indicate
certain field strengths as the field in the main solenoid was varied from
zero upward. The six field strengths thus marked off are: o, 8.34, 16.67,
25.01, 33.35, 41.68 gauss, respectively. The amount of deflection for
each particular field strength will be the distance from a base line passing
through the position for zero field to the various ploints of intersection
of the curve and the different vertical lines. These points, when plotted,
give exceptionally smooth curves, as shown in previous work.^ The fact
that the zero points for the various graphs on the same filni do not lie on
a horizontal line simply means that the camera height was readjusted
from time to time. This was necessary, because in the demagnetization
process, the mirror was sometimes displaced. It does not mean that in
the demagnetization the rods did not return to initial conditions. Careful
tests found them returning to initial conditions.

Substances possessing an initial lengthening as steel does in the Joule
effect also show a *' Villari* reversal point." There is an intimate relation
between these two phenomena. Most of the investigators* have not
found a '* Villari reversal point" in nickel. Heydweiler,^ however, found
such a reversal point in nickel. This present investigation would seem
to indicate that the presence of the auxiliary fields, either due to tempor-
ary or permanent magnetization, might have something to do with the
presence or absence of the "Villari reversal point." In a subsequent
paper will be shown how a " Villari reversal point" may be found in nickel
and how it may not.

Having shown that it was possible to get an initial lengthening in
nickel by the presence of either a remanent magnetic field in the rod itself
or by an auxiliary field outside of the rod it was of interest to study the
effect of these auxiliary fields upon substances like steel which do show ah
initial lengthening. In graphs /, /, K, L, M, N, O and P, Fig. 3, are
shown the effects of an auxiliary field upon a steel rod, described as No. 2
in previous papers.^ In /, /, K and L the rod is in an auxiliary field of
70.85 gauss and which is directed upward. This field was kept on con-
tinuously while /, /, K and L were run off. Demagnetization occurred

» Phys. Rbv., Vol. 32. p. 293, 191 1.

* Thomson, Application of Dynamics to Phys. and Chem., pp. 41-59.

* Ewing, Mag. Ind. in Iron and Other Metals, pp. 199 and 224.

* Heydweiler, Wied, Annal., Vol. 52, p. 288, 1894.
» Phys. Rkv.. Vol. 34, p. 261, Apr., 1912.

136 5. R, WILLIAMS, [

before each group. / and K are for fields up in the main solenoid while
/ and L are for fields downward. Graphs M^ N, O and P are similar
to /, /, K and L, only the auxiliary field is now directed downward. Af
and O are for fields directed upward and N and P for fields downward in
the main coil. Graphs were made, but not shown here, in which the
change in length of steel rods was studied when, without demagnetization,
the main field was applied alternately first in one direction and then in
the other. Very little effect was found due to remanent magnetism.
The earth's field also plays a negligible role in the case of steel. It will be
seen, however, that it is possible to obtain a curve for steel which shows
a shortening for all field strengths in the main coil just as a normal nickel
specimen would, the condition being that the direction of the field in
the main coil is the same as that of the auxiliary field. This last amounts
to a study of change in length of a steel rod after it has reached the point
of maximum elongation and consequently will shorten for all field
strengths which might thereafter be applied to it. I have just shown,
however, that for nickel we get the same effects from auxiliary fields
outside of the rod as from inside auxiliary fields due to remanent mag-
netism, stresses, etc. This is saying that due to causes inherent in the
specimen we might have even a steel rod behaving very much as a normal
nickel rod, i. e., shortening for all field strengths. In fact, rod no. i,
described in a former paper,^ was almost in this condition and only by
the greatest care could the initial lengthening be detected. We have
seen in the graphs already shown that nickel c£tn be made to behave either
normally or like steel in its change of length due to a magnetic field. It
seems probable, therefore, that by obtaining certain initial conditions
in the rods it would be possible to obtain the same type of change in
length for all ferromagnetic substances when subjected to a magnetic
field.

In a paper* describing a model of the elementary magnet I pointed out
that a model could be made to exhibit all of the various types of changes
in length which we find in the Joule effect in steel, nickel, cobalt, etc.,
and that the type of change in length depended upon the intrinsic field
of the elementary magnet and its initial orientation in the field imposed
upon it. In the photographs which have just been shown it has been
made dear that the type of change in length for the various substances
depended upon the character of the auxiliary fields present. These
might be either a field due to remanent magnetism or one produced by
an auxiliary coil or any other cause. In other words, these auxiliary fields

» Phys. Rbv.. Vol. 34, p. 261, Apr.. 1912.
* Phys. Rev., Vol. 34, p. 40, Jan., 1912.

No*^] COMPARATIVE STUDIES OP MAGNETIC PHENOMENA, 1 37

condition the initial orientation of the elementary magnets in the specimen
and so the type of change in length. We may now ask why under normal
conditions is the change in length in steel different from that in nickel,
cobalt, or the Heusler alloys, etc. It would seem to the writer that the
above experiments indicate the possibility of the intrinsic fields of the
elementary magnets themselves being sufficient to give conditions which
shall determine the initial orientation of the elementary magnets. As
these intrinsic fields are different for different substances, different initial
conditions will be given and so various types of changes in length will
occur as we actually find them doing. I have shown that in the case of
rods the initial orientation might be greatiy influenced by rolling or draw-
ing the specimens. In the case of steel, for instance, the elementary
magnets will line themselves up in their own magnetic fields in a definite
way and under the same conditions of previous heat treatments, states
of strain, etc., there will always occur the same t3rpe of change in length.
For a given mechanical property there is always associated a definite
magnetic property.^ This type of change in length we can alter, however,
by heat* treatment, by stresses,* and, as has just been shown, by other
auxiliary fields. In nickel or some other magnetic substance the fields
of the elementary magnets will be different and therefore the alignment
of the elementary magnets in their position of equilibrium will be different
from that of steel and so a different Joule effect. This is just an extension
of the idea of Ewing^ concerning the mutual effect of the elementary
magnets upon each other.

To return once more to the comparison of the Joule and Wiedemann
effects. In Fig. 5, i?, 5, T, U, F, tT, X and Y are shown the twists pro-
duced in the nickel rod no. 5, when subjected to a circular field due to a
current of three amperes flowing through it and with a longitudinal field
varying from zero upward. An auxiliary field was applied to annul the
vertical component of the earth's field, hence the graphs in Fig. 5 are to
be compared with £, F, G and H in Fig. 4, in which the Joule effect is
studied under similar conditions of auxiliary fields.

The graphs for the Wiedemann effect show that even with the earth's
magnetic field compensated the twist to right and left with reversal of
the main field is not symmetrical. This is undoubtedly caused by the
special effect of twist due to a longitudinal field, attention to which was
called in a former paper,^ and which is brought about by the structure
of the rod being set in a permanent twist in the drawing or rolling process.

> Journal of Cleveland Engineering Soc., Jan., 191 7, p. 183.

* Nagaoka and Honda, Phil. Mag., p. 51, Vol. 4, 1903.

* Honda and Shimizu, Phil. Mag., p. 342, Vol. 4, 1902.

* Ewing, Mag. Ind. in Iron, 171, p. 299.

* Amer. Jour. Sci., Vol. 36, p. 555, I9i3-

138 5. R, WILLIAMS. [^SS

The films for Fig. 5 are to be thought of as moving upward behind a
horizontal slit on which a spot of light is thrown from a mirror attached
to the rod. As the rod twists, the spot of light will trace on the moving
film the curves as shown. If the twist is toward the left edge of film,
it means that the lower end of the rod is twisted clockwise, as viewed
from the upper end of the rod Graphs i?, 5, T and U are for the con-
ditions that a current of 3 amperes flows upward in the rod while alter-
nately a field up and then down is applied by the main coil. R and T
are for fields directed downward and S and U for fields directed upward.
Graphs F, W, X and Y have a current of three amperes flowing downward
in the rod, while V and X are for the magnetizing force upward and W
and Y for the main field downward. Demagnetization occurred before
each graph.

These photographs of the Joule and Wiedemann effects again show that
in the Wiedemann effect there is a maximum twist for which there is no
corresponding maximum elongation in the Joule effect, the reason for
which has already been given.

Inasmuch as the graphs showing the twist due to a longitudinal field
in the paper just cited were taken without the precaution of compensating
for the vertical component of the earth's field, graphs are herewith shown
in which this precaution was taken. In M, N, R and 5, Fig. 5, is shown
a cycle of twists for varying field strengths. M and R are for fields di-
rected upward while N and S show the twist when the magnetizing force
is directed downward. Lines of twist in the structure of the rod, when
viewed from the upper end, is like a right-handed screw. A comparison
of Mi N, R and S in Fig. 5, with 22, 5, T and U in the same figure will
quickly disclose the relation which this latter effect has with the Wiede-
mann effect.

The solenoid used in this work was a new one and has been described

in a recent paper.^

Summary.

1. This paper confirms the viewpoint that primarily the Wiedemann
effect is a special case of the Joule, but that one condition prevails in the
former which does not in the latter; viz., there is a constantly changing
direction of the resultant field imposed. This condition gives rise to
several variants already noted.

2. There has been offered in this paper an explanation why the maxi-
mum twist in steel for the Widemann effect comes at lower field strengths
than does the maximum elongation in the Joule effect. In a paper by
Knott,' he speaks of this as an inexplicable fact.

* Jour. Frank. Inst., Sept.. 1916, Vol. 182, p. 353.

• Knott, Trans. Roy. Soc. Edin., Vol. 39. ?• 377. 1890.

Xo*!^] COMPARATIVE STUDIES OF MAGNETIC PHENOMENA, 1 39

3. The importance of considering the effects of auxiliary fields upon the
magnetostrictive effects has been brought out. In the case of iron, the
earth's field as an auxiliary field has very little influence on the magneto-
strictive effects as compared with those in nickel.

4. A possible explanation of why the magnetostrictive effects vary in
different substances has been presented.

Physical Laboratory,

Obbrlin Collbgb, Obbrlin, Ohio.

140 D. L. RICH. I

OSCILLATORY SPARK DISCHARGES BETWEEN UNLIKE

METALS.

By D. L. Rich.

Introduction.

TT is known by physicists that there are some twenty factors affecting
-'• the production of an oscillatory spark discharge; furthermore, with
a few exceptions, there is general agreement as to the part each one of
these factors plays. One of the exceptions noted is the effect of the elec-
trode material itself, and that particular phase of the subject forms the
basis of this paper.

In view of the very great amount of work that has been done in con-
nection with the electric spark, the effect of the electrode material is
very rarely mentioned. In practically .all research work on spark dis-
charges the spark has been formed between electrodes of like material,
so that opportunities for observing the effect of dissimilar materials have
been in general absent. Considering the extensive literature on the oscil-
latory spark discharge, the lack of experimental data for sparks between
electrodes of unlike material is surprising. On the other hand, the effect
of the electrode material on the electric arc, both D.C. and A.C., and the
behavior of what are known as crystal rectifiers, have been subjects of
wide investigation in recent years, and the well-known results therein
determined would naturally lead one to believe that the electrode material
is not a negligible factor in any electric discharge.

Previous Work on this Problem.

One of the earliest investigators of the effect of the electrode material
was Righi.^ He found no difference in the sparking potential in the case
of C, Bi, Zn, Sn, Pb and Cu. In 1878, De La Rue and Muller,* in addition
to their work on the effect of the shape of the electrodes, used their 10,000-
cell silver chloride battery to produce sparks between similarly shaped
electrodes of unlike material, Cu, Ag, Pt, Mg, Zn, Al, brass and steel.
They also found no difference, except in the case of aluminum, from which
sparks could be drawn apparently a little more easily. In 1892, Peace*

> Righi, Nuovo Cimento, 16, p. 97, 1876.

* De La Rue and Muller, Phil. Trans., Part i, p. 55. 1878.

* Peace* Proc. Soc. Lond., 52, p. 109, 1892.

VOL.X.

No.

,f] OSCILLATORY SPARK DISCHARGES. 14!

worked with Cu, Zn and brass electrodes. His method was to apply
a storage battery to two spark gaps connected in parallel. These spark
gaps were set at unequal lengths, and enclosed in separate receivers from
which the air could be exhausted; then by changing the gas pressure the
sparks were shifted from one gap to the other. No variation produced by
interchanging the electrode materials was observed. In 1903, Carr*
measured the breakdown potential difference for brass, Fe, Zn and Al.
He likewise found no difference.

When the spark gap is exceedingly short, as in the case of the coherer,
the material undoubtedly affects the discharge. Guthe* was the first
to point this out, showing in 1901 that the cohering effect between elec-
trodes of unlike material takes place at a lower voltage in one direction
than in the reverse direction. Hobbs,* using like electrodes and very
short spark-lengths (two to six wave-lengths of light) claims that the
metallic ions take part in the discharge, with the result that the material
of which the electrode is composed exerts an important influence on the
spark potential. Almy^ attempts to connect cathode fall with spark
potential. To quote him, ''The fact that different metals show marked
difference in the so-called cathode fall obviously leads to the inference that
spark potentials must to a certain extent depend on the material of the
electrode used." Later he says, "In air the cathode fall of the different
metals differs so little that it hardly seemed probable a difference in spark
potentials would be detected." His experimental work was done almost
entirely with hydrogen as the gas in which the electrodes were immersed.
He used a storage battery to furnish the voltage, and a Weston voltmeter
to measure it. In hydrogen he finds fairly consistent differences due
solely to the material of the electrodes. In air the only variation in
sparking potential that he mentions is in the case of Pt-Al electrodes,
the Pt"Al+ discharge requiring eight volts more than the Pt+Al" dis-
charge.

The work of Schuster and Hemsalech,* followed by that of Milner* and
of Royds,^ throws much light on the behavior of metallic ions in the spark
gap. These men photographed the spark on a rapidly revolving film,
allowing the light to pass through a spectroscope placed between the spark
gap and the film. The appearance of the photographed lines enabled

> Carr, Phil. Tranfl., A, 3oi, p. 419* 1903.

* Guthe, Annalen der Physik, 4, p. 762, 1901.
*Hobb8, Phil. Mag.. 10, p. 619. 1905.

^ Almy, Univ. of Nebraska Studies. Vol. 6. No. 4, 1910.

* Schuster and Hemsalech, Phii. Trans., A, 193, p. 189, 1899.
*MiIner, Phil. Trans., A, 209, p. 71, 1909.

' Royds, Phil. Mag., 19, p. 285. 1910.

142 D. L. RICH, \

them not only to identify the metallic vapors present in the spark, but
also to compute the velocities with which these vapors traveled out into
the spark gap from each electrode. Their results show that the different
metal vapors travel with unequal velocities. Since their work was
mainly the determination of ionic velocities they were not concerned with
the critical sparking potential, nor with rectification effects. It is how-
ever only a logical probability that if the metal ions travel with unequal
velocities these same ions are liberated from the electrodes with unequal
facility, and therefore the spark might start more readily from some
metals than from others; in other words, the measurements of Schuster
and Hemsalech rather support the idea that rectification effects due to
electrode material do exist.

The Problem.

If there is a rectification effect of this nature, it should manifest itself
when an alternating electromotive force of sufficient magnitude is applied
to unlike electrodes. And the behavior of the spark discharge is probably
best studied by photographing the spark gap while the discharge is taking
place.

Our problem then is really this: Apply an alternating electromotive force
to an oscillatory circuit containing a spark gap made of electrodes mechanic-
ally alike but chemically different and determine by photographing the spark
whether or not the oscillatory discharge starts as readily when one electrode
is anode as when the other is anode. During one of the series of half-cycles,
say that series consisting of the first, third, fifth, seventh, etc., half-cycles,
one of the electrodes will be initially an anode; while in the other alternate
series, consisting of the second, fourth, sixth, etc., half-cycles, this same
electrode will be initially the cathode.

To photograph the spark gap in air at ordinary atmospheric pressure,
three principal methods are available: (a) To insert either a rotating
mirror or a rotating lens between the spark gap and the stationary sen-
sitized surface of the camera. Feddersen, and Trowbridge, for example,
used the rotating mirror, while Boyd used a series of rotating lenses.

(b) To separate the otherwise superimposed oscillations by blowing the
sparks from the narrow to the wide end of a V-shaped spark gap by means
of a powerful blast of air. Klingelfuss was the first to use this method.

(c) To receive the image of the spark on a rapidly revolving plate or film.
Pierce, and Lodge and Glazebrook used a revolving plate, while Schuster
and Hemsalech used a revolving film.

The second method was discarded as undesirable, and throughout this
work there has been used a combination of the first and the third methods

L Review. Vol. IX., Seconc
August, 1917.

Fig. 6-
D. L. RICH.

uS^^'] OSCILLATORY SPARK DISCHARGES. 1 43

outlined above, the photographic lens being moved With rather slow
velocity, and the photographic film or plate with a much higher velocity.

First Method Used — Plate Camera.

In order to observe the relative number of spark trains per half-cycle
no excessive speed of the photographic surface is necessary. For this the
following method was found. satisfactory. An ordinary glass plate nega-
tive, held in its customary plate holder which in turn was clamped cen-
trally on the end of the shaft of a small motor, and at right angles to the
shaft, so that the plate could be turned at a fairly high speed in its own
plane, was the arrangement used for moving the photographic surface.
In order to prevent the superposition of images the lens of the camera was
swung slowly across in front of the plate and parallel to it, so that the
successive spark trains traced out a spiral on the plate.

To produce these sparks a spark gap, an inductive resistance, and a
condenser were connected in series, and then the condenser put directly
across the secondary of a rather leaky high voltage transformer whose
primary was connected to no- volt 60-cycle mains.

Figs. I, 2 and 3 are reproductions of photos secured in this way. Each
spot represents not a single spark but a complete train of sparks, the
speed of rotation not being sufficiently great to separate the individual
oscillations. The groups of spark trains correspond to the half-cycles
of the exciting primary current. The asynunetry of the spark trains
per half-cycle is very plainly evident, every second half-cycle containing
many trains, and the alternate half-cycles fewer trains. In Fig. i is a
half-cycle (marked "o") that produced no spark trains whatsoever.
In Fig. 2 several alternate half-cycles trail off into a tail, indicating that
the oscillatory discharge degenerated into an ordinary arc discharge.
To test this, the capacity and the inductance were disconnected, and Fig. 3
is the appearance of the resulting arc.

Interpretation and Predictions.

Figs. I, 2 and 3 are photographs taken when the electrode materials
were copper and iron. The asymmetry of the spark trains per half-cycle
indicates that there is a rectification effect present in the oscillatory dis-
charge between two electrodes of dissimilar material. Probably if the
critical voltage were secured (a difficult task, but not impossible) spark
trains would be produced only on alternate half-cycles, the others being
entirely suppressed. Even if this critical voltage were not used, but
instead a voltage somewhat higher than the critical, rectification effects
when present would manifest themselves in at least three ways:

F«^

144 D. L. RICH.

1 . In the ReUUwe Number of Spark Trains per Half -cycle. — If we assume
that the ^)arkB start more easily at one electrode than at the other, we
are led immediately to the conclusion that the sparks must start more
readily on one half-cycle than on another; that one set of alternate half-
cycles will produce more spark trains than the other alternate set. The
following diagram (Fig. 4), it is reasonable to suppose, might represent
what our assumption leads us to
expect. Suppose the electrodes
are Cu and Fe, and that the
first half-cycle occurs when Fe is
initially positive, and the second
half-cycle when the electrode Fe
is initially negative; and suppose ©•- o»*
further that we assume that the Fig. 4.

spark starts the more readily

when Fe is negative than it does when Cu is negative. This amounts to
the same as assuming that the sparking potential, or the difference of
potential necessary to initiate a spark, is relatively high on the Cu~Fe+
half-cycles and relatively low on the Cu+Fe~ half-cycles. The potential
builds up along the curve AB until it reaches the sparking potential CD
at the point E. Then a discharge takes place, either unidirectional or
oscillatory as the case may be, with the result that the potential difference
is reduced to a magnitude less than that necessary to maintain the dis-
charge (not necessarily reduced to zero, however). The voltage then
builds up again, let us say along some such line as FG, and the process
repeats itself until finally the curve reaches H and no further discharge
takes place until the potential difference builds up in the opposite sense
in the next half-cycle, with Fe initially negative. Owing to the relatively
high potential necessary to initiate a spark, only a relatively small number
of grains will be produced in this Cu~Fe+ half-cycle.

During the next half-cycle, if the sparking potential // is small, the
discharges will have a chance to begin earlier in the half-cycle; after the
falling off of the potential due to the discharge, the potential can build
up again to the necessary sparking magnitude more quickly; and the
sparks can start and last later in the half-cycle; all three of these factors
tend to produce more discharges in this Cu'*"Fe"' half-cycle than in the
preceding Cu~Fe+ half-cycle.

2. In the Relative Number of Individual Oscillation Sparks per Train. —
When a spark does occur at the higher sparking potential, this spark
might reasonably be expected to be of a more violent nature than a spark
produced at a lower sparking potential. If more violent, not only would

vS^a^] OSCILLATORY SPARK DISCHARGES. 1 45

the condenser be more strongly charged in the opposite sense, but there
would also be more ions produced in the spark gap, with the result that
the spark gap resistance would be decreased. Both the stronger charge
and the lowered resistance would lead one to expect that the oscillations
following the initial spark would be more numerous. The large initial
charge and the smaller damping decrement would each tend to prolong
the duration of the wave train.

3. In the Relative Number of Spark Trains Containing an Even Number ^
or an Odd Number, of Individual Oscillation Sparks. — ^So far as the
material affects matters, if the initial spark in any train starts with ease,
the third, fifth, etc. (the odd sparks), in that same train, since they start
from the same electrode, should also start with ease, with greater ease
than the second, fourth, etc. (the even sparks), which originate at the
other electrode, from which the spark starts with difficulty. It would
naturally be expected that any spark train would stop with an easy spark,
i. e., at the beginning of a spark difficult to start. That is, if the initial
spark of any train starts easily, that particular spark train might be
expected to contain ah odd number of sparks. Likewise, the train whose
first spark originates at an electrode from which the spark starts with
difficulty should in general contain an even number of individual sparks.

But in addition to the material of the electrodes there are at least two
other factors, (even with the electrodes mechanically and chemically
alike) that probably have an influence on the oddness or evenness of the
number of sparks per train. The primary current is changing while the
secondary discharge is taking place. The flux change in the secondary
due to this slowly changing low frequency primary current, combined
with the flux change in the secondary due to the rapidly changing high
frequency secondary current, since these two changes are in the same sense
during half the sparks and in opposite sense during the other half, would
probably produce an asymmetry that would favor one set of sparks always
whether the electrodes were of the same material or not.

Also, the ionization produced in the spark gap is a function of the
velocity of the ions, which velocity is in turn dependent on the potential
at which the ions were emitted, — a high potential difference producing a
high velocity and therefore a low capacity for ionization. So considering
the three factors, and not knowing their relative magnitudes it is impos-
sible to predict which spark trains will contain an even number of in-
dividual oscillation sparks, and which an odd. number.

Reasons for Changing to a Film Camera.

To investigate the above matters it became necessary to rearrange
both the camera and the oscillatory circuit. The first method used, and

146 D. L. RICH. @SSS

the photographs secured, samples of which are shown above, are satis-
factory to determine the relative number of spark trains per half-cycle;
bat to obtain photographs showing individual oscillation sparks the pho-
tographic surface must be made to move at a very much higher rate, and
the period of the oscillatory discharge must be lengthened to secure a
much lower frequency.

To lower the frequency a larger inductance coil and a larger condenser
were built. The inductive resistance was a circular coil a meter in
diameter and a meter high, and consisted of about a hundred turns of
no. 12 coppered iron wire wound on a rough wooden frame. Its low
frequency resistance was 6.5 ohms, and its low frequency inductance
25 millihenrys. The condenser was made of 48 large panes of ordinary
window glass, each pane shellacked, coated on both sides to within five
centimeters of the edge with tin foil, then given two more coats of shellac.
The panes were mounted in a wooden frame with air insulation. When
thoroughly dry the leakage was not excessive, but the absorption was very
bad. The low frequency capacity ranged from an eighth to a half micro-
farad, depending on the duration of the charge and of the discharge.

To increase the speed of the camera the glass plate was abandoned and
a film used instead (Eastman extra rapid speed film, about 55 inches long
and 2^ inches wide). The film was wrapped around the flat outside
rim of a wheel 40 cm. in diameter. This made the total exposed length
of the film about 1,260 mm. The wheel was mounted on a motor running
2,000 revolutions per minute, giving a linear velocity to the film of about
42,000 mm. per second. In order to hold the film in position at this
rather high speed it was found necessary to bolt the film to the wheel
with twelve bolts arranged zigzag around the circumference. By actual
measurement of the developed films it was found that this inductance,
capacity and film speed resulted in a separation of the individual oscilla-
tion sparks to distance of 4.1 mm., a distance amply 3uflicient for all pres-
ent purposes.

As before, exposure was made by swinging the lens across the film
while the film was rotating at a high speed, so that the spots of light
traced out a continuous spiral on the film. The lens was moved by a
heavy weight which in falling from ceiling to floor picked up counter-
balancing weights so that its speed was kept approximately constant.
Further the weight in falling automatically operated a mercury switch so
that the spark discharge took place only while the lens was passing in
front of the film. By adjustment it was found possible to cause the spiral
to trace out twenty complete turns on each film, thus enabling the spark
gap to be kept under continuous observation for over half a second at a time.

VOL.X.1

Naa. J

OSCILLATORY SPARK DISCHARGES.

showing the groups of spark trains in over sixty consecutive half -cycles all
spread out in a single line photograph over twenty-five thousand millimeters
long.

Data Secured from a Typical CuFe Photo.

Half.cyelaa in Which Pa Waa InitiaUy

Ntfotrve,

Half.

•cycles in Which Pe Was Initially
PoMttivt.

Half-

eyde.

Num-
ber of
Spark
Traina,

Number of Sparka in Bach
Train.

Total
Num-
ber of
Sparka.

Half,
cycle.

Num-
ber of
Spark
Trains.

Number of Sparks in
Bach Train.

Total
Num-
ber of
Sparks.

Incomplete

5 2 2 2

5 3 3

4 2 2 2 2 2 2

5 3

4 2 3 3

4 2 2 2 2 2

5 3

4 2 2 2 2 2

5 3

4 2 2 2 3

4 3 3

5 2 2 2 3

5 3 3

422222222

5 3

42222222

5 3 3 3

4 2 2 2 2 2 2

5 3 3

4 2 2 2 2

4 3 3

4 12 3 2

4 3 3

4 3 2 3

4 3 3

4 2 2 2 2*2 2 2

4 3 2 3

4 2 2 2 2 2

5 3 3

422222222

5 3

42222222

5 3 3

4 2 2 2 2 2 2

3 3 3

3 3 2 2 2

3 2 2 2 3

422222222

3 3

4 12 2 2 2 2

3 3 3 3

3 2 2 2 2 2 2

3 3 3

4 2 2 2 2 2 2

3 3 3 3

4 2 2 2 2 2 2

5 3

422222222

4 2 2 3

4 2 2 12 2 2 2

3 3 3

42222222

4 2 2 3

22222222222

3 3 3

32222222

3 3 3 3

42222222

3 3 3 4

Incomplete

3 3 3 3

Incomplete

3 3 3

To show relative number of spark trains per half-cycle, and also to
show relative number of oscillation sparks per train, it is desirable to use
a short spark gap, and highly damped oscillations. Fortunately these
conditions are the very easiest possible.

148 D, L, RICH.

A large number of photographs were taken, in an interval extending H

over two years. Most of the work was done on CuCu and CuFe elec- Z

trodes in an attempt to settle definitely the point in question with these ;;

particular metals. Later electrodes of zinc and bismuth were used in ^

various combinations with each other and with copper and iron. Many ^
different specimens were used, and several different shapes. The sparks

were always produced in air at atmospheric pressure. The spark gap ^

was varied in length from o.i mm. to 3 or 4 nun., while the lens of the ••

camera was so placed that the image of the spark gap was slightly longer Z-

than the spark gap itself. ^

Figs. 5 and 6 are sections of typical films showing the appearance of '^

the sparks between CuFe electrodes and CuCu electrodes respectively. ^
Owing to the fact that some of the beginning and the ending half-cycles

are incomplete on the edges of the films, the number of half-cycles ob- -^

servable in the two series are in general unequal; however, no error is ^

thus introduced as average values are desired, and any number of half- ^

cycles may be averaged. ^

CtiFe Summary, ^

Cii+Fr-, Cu-F«*, ^

Total number of half-cycles observed 30 32 ^

Total number of spark trains 205 101 *^

Average number of spark trains per half -cycle .6.2 3.16 ^

Total number of oscillation sparks 472 327 ^

Average number of oscillation sparks per train 2.3 3.23 ^

Number of trains consisting of 1 spark 3 0

Number of trains consisting of 2 sparks 166 9 ''

Number of trains consisting of 3 sparks 10 71

Number of trains consisting of 4 sparks 23 9 ^

Number of trains consisting of 5 sparks 3 12 ^

Total number of spark trains consisting of an even number y

of sparks 189 18

Total number of trains consbting of an odd number of

sparks 16 83

Per cent, even 92.2 17.7 ^

Per cent, odd , 7.8 82.3 ^

An examination of the CuFe data summary shows that the spark trains / '

on the Cu+Fe"* half-cycles were almost double the number of trains on /

the Cu~Fe+ half-cycles (6.2 to 3.16); that the number of oscillation ^
sparks per train was over a third larger (2.3 to 3.23) in the Cu"Fe+ half-
cycles than in the Cu+Fe" half-cycles; and that in the Cu""Fe+ half-cycles
82 per cent, of the trains contained an odd niunber of sparks, while in the
Cu+Fe* half-cycles 92 per cent, of the trains contained an eoen number of
sparks.

•^

nsi"a^l oscillatory spark discharges. 1 49

Interpretation.

The relative number of spark trains per half-cycle, and the relative
number of sparks per train indicate clearly that the Cu+Fe" discharge
takes place more readily than the Cu~Fe+ discharge. The fact that an
odd number of sparks per train predominates during the Cu"Fe+ half-
cycles and an even number during the Cu+Fe"" half-cycles indicates that
the ionizing effect is the predominating influence in determining the
evenness or oddness, as the sparks as a rule stop with the higher voltage,
higher velocity, lower ionizing discharge.

In contrast with the preceding data, which was secured from a typical
series of discharges between electrodes mechanically alike but chemically
different, and which show rectification effects attributable solely to elec-
trode material, compare the following set of data, from a characteristic
series of discharges between CuCu electrodes.

Summary of Data Secured from a Typical CuCu Photo,

Odd Bvto

Hftlf-cycl«t. Half-cycles.

Total number of half-cycles observed 31 33

Total number of spark trains 63 74

Average number of spark trains per half -cycle 2.03 2.24

Total number of oscillation sparks 250 291

Average number of oscillation sparks per train 3.97 3.93

Number of trains consisting of 1 spark 0 0

Number of trains consisting of 2 sparks 0 0

Number of trains consisting of 3 sparks 10 7

Number of trains consisting of 4 sparks * ... 45 64

Number of trains consisting of 5 sparks 8 3

Number of trains consisting of an «ofn number of sparks . 45 64

Number of trains consisting of an odd number of sparks .18 10

Per cent, even 71.5 86.5

Per cent, odd 28.5 13.5

An examination of the CuCu data summary preceding shows that the
average number of spark trains per half-cycle is practically the same in
the two series (2.03 to 2.24) ; that the average number of individual sparks
per train is almost identically the same (3.97 to 3.93) ; and that in both
series the spark trains are predominantly of four sparks each. In not
one of these three respects is there even any hint of rectification effects.
The irregularities which do occur in discharges of this nature are prob-
ably due to the conducting variations in the spark gap, possibly caused by
air-currents, etc., variations which prevent the absolutely uniform charg-
ing of the condenser.

In every photograph, without a single exception, I have found the CuCu
discharge to be symmetrical, and the Cu^Fe~ discharge to be more easily
produced than the CwFe^ discharge.

1 50 D. L, RICH.

Determination of Polarity.

In order to determine which particular kind of spark should be asso-
ciated with the Cu^Fe"*" discharge, and which with the Cu+Fe*", the
A.C. primary circuit was disconnected and there was substituted a 220-
volt D.C. source, through a suitable non-inductive high resistance and a
knife switch. The fact that the primary E.M.F. was high, produced a
rapid change of flux in the transformer when the switch was closed, and
consequently a vivid spark in the secondary circuit; yet the resistance
inserted in the primary circuit resulted in less than a volt across the
primary coils, so that it was perfectly safe to put an ordinary 250-volt
voltmeter directly across the secondary of the high potential transformer,
and thus .determine definitely and easily the polarity of the spark elec-
trodes at make and at break. The voltmeter behaved as a ballistic gal-
vanometer, and by the direction of its "kick" gave positive evidence
concerning the polarity.

The camera film was set in motion, the camera shutter closed except
at "make," and a series of make spark photos secured. Then the lens
was displaced sideways slightly, the shutter closed except at "break,''
and a series of break spark photos obtained. The electrodes were then
reversed, and the make and the break photos again taken, all on the same
film, side by side.

When the primary circuit was closed, a single spark train was expected.
As a matter of fact the photographs showed several spark trains for each
make, the number varyiftg from four to forty-two, depending on the
length of the spark gap. This may be explained as follows:

Suppose the total quantity of electricity flowing into the condenser
of the secondary circuit be plotted against time, giving the familiar curve
of Fig. 7. A quantity qi might be sufficient to charge the condenser to

- 1WV\^,

Fig. 7. Fig. 8.

the sparking difference of potential, and an oscillatory discharge would
then occur. A further quantity qt — qi might again charge the con-
denser, and another oscillatory discharge ensue; and again a quantity
qi — qt, and still further ^4 ^ qtt and so on, might each cause a spark
train. The fact that at each make the first spark trains were not only

JJ^j^] OSCILLATORY SPARK DISCHARGES, 151

more intense but also made up of from five to eight oscillation sparks
each, while the later spark trains were of less intensity and also of shorter
length (2, 3, or 4 oscillation sparks each) supports the above explanation.
The curve (Fig. 8) probably represents what took place:

Owing to the arcing effect at the knife switch when the primary circuit
was opened, usually only one spark train was observed at "break," and
it was usually much fainter than the sparks at "make."

Several films were used in this manner. In every case the photos of the
Cu""Fe+ discharge consisted of light, narrow, faint spots alternating with
heavy, broad, darker spots, and always beginning with the fainter spot,
as shown in (a) Fig. 9; while the Cu+Fe" discharge photo was of similar
alternations, but always beginning with the broad dark
spot, as shown in (6), Fig. 9. The metallic vapor liber-
ated in the spark gap when iron was anode always pro- * ■"—

duced the much more intense light effect. On all the «

photos, whether produced by direct current or alternat- pj- 9

ing current means, the polarity of the electrodes could
be identified easily and positively. Furthermore, on the D.C. photos,
the Cu+Fe" discharge without exception consisted of an even number of
oscillation sparks; while the Cu"Fe+ generally, though not always, con-
sisted of an odd number of sparks.

When the electrodes were of the same material, for example CuCu,
the sparks were always of like character throughout, gradually growing
fainter as the amplitude decreased, as shown in (c). Fig. 9. Furthermore
the number of individual oscillation sparks when the electrodes were
alike was predominantly even.

Further Results with Other Electrode Combinations.

It next seemed desirable to investigate the behavior of some other
metals when used as electrodes, in order to see whether or not there exists
a consistent rectification series among conductors in general. The same
method was continued, and zinc, bismuth, copper and iron electrodes
were used repeatedly in all possible combinations.

Iron-bismuth was one of the first combinations tried. Below is a
sununary secured from a typical Fe-Bi photo.

An examination of this summary, particularly the relative number of
spark trains per half-cycle, and the relative number of oscillation sparks
per train, indicates that the discharge can start more readily when iron
is negative. E^ch of the other films of this particular electrode combina-
tion indicated the same rectification. Not much reliance, however, can
be placed on the relative number of even and odd spark trains.

152 D.L. RICH. ^^

Iron-Bismuth Summary.

F«-Bi+. Fe+Bl-.

Total number of half -cycles observed 28 29

Total number of spark trains 146 83

Average number of spark trains per half -cycle 5.21 2.86

Total number of oscillation sparks 218 184

Average number of oscillation sparks per train 1.49 2.21

Number of trains consisting of 1 spark 76 15

Number of trains consisting of 2 sparks 68 35

Number of trains consisting of 3 sparks 2 33

Number of trains consisting of an even number of in-
dividual sparks 68 35

Number of trains consisting of an odd number of in-
dividual sparks 78 48

Per cent, even 46.5 42.1

Per cent, odd 53.4 57.8

After finding that the spark discharge could be initiated more readily
from iron that from either copper or bismuth (always assuming that the
discharge is electronic in nature, so that the current flows from anode to
cathode), the next step seemed to be the investigation of the copper-bis-
muth spark gap. Below is a copy of the data secured from a character-
istic CuBi film. Note the 3's.

Copper-Bismuth Summary,

Cu+BI-. CB-Bt*.

Total number of half-cycles observed 32 32

Total number of spark trains 69 34

Average number of spark trains per half -cycle 2.16 1.06

Tots^l number of oscillation sparks 201 101

Average number of oscillation sparks per train 2.91 2.97

Number of half -cycles producing no spark 1 5

Number of trains consisting of 1 spark 0 0

Number of trains consisting of 2 sparks 7 1

Number of trains consisting of 3 sparks 61 33

Number of trains consisting of 4 sparks 1 0

Number of trains consisting of an even number of in-
dividual sparks 8 1

Number of trains consisting of an odd number of in-
dividual sparks 61 33

Per cent, even 11.5 3

Per cent, odd 88.4 97

Here again, as in all of this work, the criterion as to rectification lies
in the relative number of spark trains per half-cycle. After the spark
is once started various extraneous and uncontrollable irregularities lessen
the reliability that can be placed on the way in which that particular
spark train continues. On the above film, twice as many spark trains
originated from the bismuth anode as from the copper anode. And five

f^^] OSCILLATORY SPARK DISCHARGES. 1 53

times as many failures to produce any discharge whatsoever are charged
to the copper anode.

Another combination tried was zinc and iron. The spark when iron
is anode is always easily recognized, the light from the spark then ap-
parently being exceedingly rich in actinic rays.

Iron-Zinc Summary,

Pe-Zn+. Fe+Zn-

Total number of half -cycles observed 31 29

Total number of spark trains 96 41

Average number of spark trains per half -cycle 3.1 1.41

Total number of oscillation sparks 181 115

Average number of oscillation sparks per train 1.88 2.80

Number of trains consisting of 1 spark 46 4

Number of trains consisting of 2 sparks 17 1

Number of trains consisting of 3 sparks 31 35

Number of trains consisting of 4 sparks 2 1

Number of trains consisting of 5 sparks 0 0

Number of trains consisting of an even number of in-
dividual sparks 19 2

Number of trains consisting of an odd number of in-
dividual sparks 77 39

Per cent, even 19.8 4.9

Per cent, odd 80.1 95.0

Here again the half-cycles in which iron was initially the anode pro-
duced considerably more than twice as many spark trains as the other
series. In fact» the discharge took place so readily from the iron anode
that the condenser could receive only a very small quantity of electricity,
so small that a return spark was very frequently impossible, as is shown
by the fact that 46 of the trains (48 per cent, of them) were unidirectional,
one spark discharges.

One of the most prominent cases of rectification which was observed
occurred in connection with a copper-zinc spark gap. The data for this
particular gap is inserted here not as a typical case, but as a special case,
illustrating the fact that with proper adjustment it is possible to initiate
spark trains from one electrode alone. The necessary adjustment prob-
ably would be difficult to make, and still more difficult to maintain, but
the following data show that it can be done.

Other films in the CuZn spark gap, while not so prominently asym-
metrical as this one, always showed a decided preponderance of spark
trains initiated during the half-cycles when zinc was anode, indicating
that a discharge could start from zinc much more readily than from copper.

An inspection of the foregoing summaries shows that rectification effects
manifest themselves prominently in the relative number of trains per
half-cycle, and generally in the relative number of sparks per train.

154 »• I" RICH. ^J2?

Coppef'Zinc Summary,

Ctt-Zo-^. Ctt*Zo-.

Total number of half-cycles observed 32 30

Total number of spark trains 14 143

Average number of spark trains per half-cycle 1.27* 4.77

Total number of oscillation sparks 38 253

Average number of oscillation sparks per train 2.71 1.77

Number of half-cycles producing 0 trains 21 0

Number of trains consisting of 1 spark 1 55

Number of trains consisting of 2 sparks 2 72

Number of trains consisting of 3 sparks 11 10

Number of trains consisting of 4 sparks 0 6

Number of trains consisting of an even number of in-
dividual sparks 2 78

Number of trains consisting of an odd number of in-
dividual sparks 12 65

Per cent, even 14.3 54.5

Per cent, odd 85.7 45.4

Since other factors enter into the cause of oddness or evenness of the
number of sparks per train, not so much reliance can be placed on the
relative niunber of odd and even trains.

As a result of the work discussed in this paper the conclusion is reached
that rectification effects do exist in oscillatory discharges between the
unlike metals used, the order being Fe, Bi, Zn, Cu, a^park being initiated
with the greatest facility from iron, and with the greatest difficulty from
copper.

More complete data of this nature, involving the various elements
which may be used as electrodes, may throw light on the stability of
ionic or electronic aggregations or orbits within the atom.

Comparison with a Former Result.

It has already been mentioned that Guthe was the first to suggest that
the discharge in the case of the coherer is undoubtedly affected by the
material of the electrodes. In- an attempt to measure the smallest
potential difference necessary to produce coherer action Guthe* experi-
mented with many different metals, among which were the same four,
Cu, Bi, Fe and Zn, mentioned in this paper. Since in the coherer the
two electrodes are either in actual contact or else separated by an ex-
ceedingly thin layer or film, the conditions are somewhat different from
the ordinary spark gap. Further, the potential difference applied to the
coherer was a very slowly changing, and finally constant, statical, battery
potential difference; whereas the voltage used in this work was always a
rapidly changing, alternating, transformer potential difference. Never-
theless, if the discharge is electronic, either method should yield some

> 1 1 half-cycles only.

* Guthe, Ann. d. Physik, 4, p. 762, 1901.

J52"a^] OSCILLATORY SPARK DISCHARGES, 1 55

information concerning the facility with which electrons are torn from
the anode and fired across the spark gap or the coherer film. Guthe
arranges the metals in the order Bi, Fe, Zn, Cu, the series beginning with
the easiest and ending with the hardest; i. e., ending with the metal
requiring the highest potential difference to produce coherer action.
His series differs from the series proposed in this paper only in the rel-
ative positions of bismuth and iron.

Summary.

Many photographs of the oscillatory spark discharge between electrodes
mechanically alike, but chemically different, were taken, in an attempt
to determine whether or not the material of the electrode has any in-
fluence on the initiation of the discharge.

Electrodes of copper, iron, zinc and bismuth were used; also both
alternating currents and intermittent direct currents were used; in pro-
ducing the required potential differences.

The interpretation of the relative number of spark trains per half-cycle,
the relative number of individual oscillation sparks per train, and the
relative number of trains containing odd numbers and even numbers of
individual sparks, is given, with reasons for such interpretation.

When the electrodes were alike symmetrical discharges were always
found.

When the electrodes were of two unlike metals decided rectification
effects were always produced, being very pronounced when copper was
one of the electrodes, and most prominent when iron was the other elec-
trode. In other words, the material of the electrodes is not a negligible
factor in the initiation of a spark discharge.

If the discharge is electronic, the electrons are emitted from iron more
easily than from bismuth or zinc, and much more easily than from
copper; they are emitted from bismuth more easily than from zinc or
copper; and from zinc more easily than from copper. Arranged in a
rectification series, these metals stand

Fe, Bi, Zn, Cu.

The rectification effects seemed marked and consistent throughout.
Many and various specimens of metals in various shapes were used all in
air at ordinary atmospheric pressure.

In conclusion, my thanks are due to Professor K. E. Guthe, under whose

helpful supervision this work was done; and also to Professor N. H.

Williams for much good advice during the progress of the work.

University op Michigan,
Ann Arbor, Michigan,
April I, 1915.

156 LOUIS K. OPPITZ, [ISiS

OPTICAL CONSTANTS OF THE BINARY ALLOYS OF SILVER

WITH COPPER AND PLATINUM.

By Louis K. Oppitz.

Historical Introduction.

THE first studies in the optical constants of alloys were those of
Drude^ who investigated three alloys: (a) one of 18 k. gold alloyed
with silver, copper, and a small quantity of iron; (6) one of copper-nickel;
and (c) Wood's alloy. He made no attempt, however, to study any
complete series of alloys related according to some well conceived prop-
erty.

Willi Meier* measured the optical constants of Wood's alloy and those
of an alloy of gold and silver of equal parts by weight. His main interest
was in the study of optical constants for a series of wave lengths which
extended into the ultra-violet region.

Bemouilli* measured the optical constants of a number of alloys, that
form solid solutions; but restricted his examinations to small concen-
trations, that is to dilute solid solutions. His work includes a study of
the optical constants of Ag-Tl, Ag-Sn, Cd-Hg, Cu-Sn and Cu-Ni. The
main interest in his work is his method,* which consisted of the measure-
ment of the minimum azimuth of restored polarization.

Voigt* has criticized the mathematical formula employed by Bemouilli
as an illegitimate approximation.

Littleton* was the first to study the variation of optical constants for
entire alloy series. He investigated the alloy series of Cu-Ni, Fe-Mn,
Ni-Fe, Ni-Si, Al-Cu and Cu-Fe. These alloys seem not to have been
chosen for the purpose of studying group characteristics.

In 1912, Eckhardt^ investigated a series of gold-silver alloys. Gold
and silver form an unbroken series of solid solutions. The series investi-
gated consisted of ten members of progressively varying compositions.
The concentration-refractivity curve of the series is continuous and shows

> Dnide, Ann. d. Phys.. N. F.. Vol. 39. 1890, pp. 481-554*

« WUU Meier, Ann. d. Phys., Ser. 4. Vol. 31, 1910. pp. 1017-1099.

* Beraouilli, Zeitschr. d. Elektro-chem., 15, pp. 646-648.

* Ann. d. Phys., Ser. 4, Vol. 29, pp. 585 et seq.

» Voigt, Ann. d. Phys.. Ser. 4. Vol. 29. 1909, pp. 956 et seq.

* Littleton, Phys. Rbv., Vol. 32, 191 1. pp. 453 et seq.

' Eckhardt, Doctor's Thesis, University of Pennsylvania.

NoHi^] OPTICAL CONSTANTS OF BINARY ALLOYS. 1 57

a distinct but weak maximiun, while the absorptive index curve shows a
distinct minimum at about the same concentration. The indices of
refraction of nearly all of the gold-silver allojrs are higher than those of
either component forming the series.

Object, Theory and Method of the Present Investigation.

The object of the present investigation is a study of the optical con-
stants of two complete series of binary alloys, silver-copper and platinimi-
silver.

The copper used in the alloys was electrolytic copper, while the silver
was 1,000 fine assay silver.

The alloys were approximately of the same size and mass. The masses
of the metals constituting the alloys were carefully determined on a
chemical balance. The alloys were also weighed after being fused. In
no case was there a greater loss due to evaporation than one part in about
three himdred. The boiling point of silver is about 1950® C. In order
to avoid the loss of silver by evaporation, the platinum was first fused,
and the silver was introduced gradually into the melted platinum. The
reguhis was then carefully stirred by means of a carbon rod and was kept
at red heat for several hours, to insure a homogeneous mixture.

The alloys were fused in graphite crucibles in a resistor furnace. They
are free from graphite, as is shown by the values of the optical constants
of the pure metals. The source of energy was an alternating current
passed through a step-down transformer.

Polishing of the Mirrors.

The method of polishing was approximately that of Drude. After
the alloy had cooled it was mounted and a plane face was turned on it in
a jeweler's lathe. It was then treated with emery. Fine grades of
French emery paper of four degrees of fineness (t. «., o, 00, 000, 0000)
w ere used. The process of polishing began with the use of the o grade
that being the coarsest. The specimen was stroked in a definite direction
against the emery paper. The emery paper was held on a smooth plate
of plane glass. Each mirror required individual treatment. The
pressure of the stroke was adapted to the hardness of the particular alloy.
The surface of the alloy was stroked so as to give to the scratches a single
definite direction. Then the mirror was stroked in a direction at right
angles to the scratches imparted to it by the coarsest grade of emery
paper, against an emery paper of the next grade of fineness and so on
until the finest grade of emergy paper had been used. Each succeeding
grade of emery paper thus tended to remove or to render less deep the

T58 LOUIS K. OPPITZ, [^S£

scratches introduced by the preceding, and to insure a plane surface. If
any scratches remained after the finest emery had been used, recourse
was had to a burnishing tool like that used by silversmiths. Much care
was exercised to keep the surface of the emery paper free from dust and
other forms of contamination.

Drude's criterion for a satisfactory optical surface was used: the azi-
muths of restored polarization for light parallel and perpendicular to the
scratches must be approximately equal. The phase change was found to
be invariable for a given angle of incidence so long as the mirror
remained free from surface layers.

Optical Methods.

The source of light was a Bimsen flame colored by means of fused NaCl.
This light was filtered though an aqueous solution of KtCriOr which
rendered the resulting light practically that of the D lines of sodium.
The light incident upon the surface to be studied was plane polarized
at an azimuth of 45^. This light after reflection became elliptically
polarized and was reconverted into plane polarized light by means of a
Soleil-Babinet compensator. The azimuth of restored plane polarization
was determined by means of an analyzing half-shadow-nicol system.
Then the analyzing nicol was set for extinction and the phase change
was determined by the use of the compensator. The angle at which
the plane polarized light became incident upon the surface of reflection
was carefully determined by reading the position of the telescope from the
goniometer circle. In order to determine the azimuth of restored polari-
zation, a modified form of the Zehnder^ half-shadow polarimeter was used.
This consisted of the usual analyzing nicol and a movable smoked glass
wedge, adjacent to the nicol and moving over a fixed smoked glass wedge.
In its original form the polarimeter was made up of an analyzing nicol
adjacent to a fixed smoked glass plate. The intensity of the light used
for studying the optical properties of the surfaces was found to vary for
different angles of incidence and for different optically reflecting surfaces.
It was therefore found that relatively large angles of incidence were the
most favorable. At suggestion of Dr. Eckhardt, of this laboratory, the
fixed smoked glass plate to which reference has been made was replaced
by a movable smoked glass wedge, which could be varied so as to change
the length of the path traversed by the light passing through it. This
rendered it possible to adapt the length of the path to the intensity of the
light traversing the polarimeter. This gave half shadow equality through
a range varying from 7® to 21®. The determination of the position of

> Zehnder, Ann. d. Phys., 26* 1908, pp. 985-1018.

.640

2.63

.620

2.57

No*^] OPTICAL CONSTANTS OP BINARY ALLOYS. 1 59

extinction of the analyzing nicol with the polarizer depended upon judging
half-shadow equality. Half-shadow equality is most easily judged when
the illumination through the analyzing nicol and smoked glass appears
homogeneous and intense. Two half-shadow equality positions were
viewed, one on each side of the extinction position of the analyzing
nicol. Then the analyzing nicol half-shadow device was rotated ap-
proximately i8o® and two other half-dhadow equality positions were
found. Thus,, there were four readings in all from which to find the
extinction position of the nicol. The arithmetical mean of the positions
before and after extinction gives the extinction position.

Much practice was necessary for attaining proficiency in the judgment
of half-shadow equality. After considerable preliminary practice, the
initial step in the experimental work was to determine the optical con-
stants of electrolytic copper. The experimental values obtained for pure
copper are as follows:

4.10 £>nide,
4.14 L.K.O.

The difference in the two sets of values is probably explainable on the
basis that the two specimens of copper used, differed in purity. After
determining these optical constants for pure copper, those of nine different
alloys of silver-copper, of eight alloys of silver-platinum and pure silver
and pure platinum were measured. The entire eleven points of the silver-
copper curves (Fig. 3) and the entire ten points of the platinum-silver
curves were experimentally determined (Fig. 4).

In the figures, the variation in the composition of the alloys is ex-
pressed in terms of the atomic per cent, of copper. The reflecting power
was obtained by calculation, and not by direct measurement. No ex-
planation is at present offered for the anomalously high reflecting power
of the silver-copper alloy of 4.99 per cent, concentration.

Working Formulae.

The well known formulae of Drude were in the calculation of the optical

constants:

n*(i + IP) = tan*P sin* 0 tan* 0. (i)

The atomic per cent, of one component is given by

loop

* ~a'

p + iioo-p)-^

l6o LOUIS K. OPPITZ. Ess

where p — per cent, by weight of this component, a = its atomic weight
and b ~ the atomic weight of the other component.

jr - tan Q (2)

tan 4 = sin 0 tan 2 i* (3)

cos 2^p = cos Q an 3P (4)
„ _ n'(i + JP) + I - 2»

where

B»Ci + JO) + I + a«

ff = the index of refraction,
K = the absorptive index,

4 = the phase change,

<f, B the azimuth of restored polarization,
R — the reflecting power,

^ — the angle of incidence.

Experimental Results.
Silver-Copper Alloys.
Silver and copper form a series of alloys in which there are two limited
series of solid solutions, separated by a gap. This gap consists of a series
of eutectiferous alloys. As one
withdraws from pure silver, silver
crystals, ». «., crystal type I. sepa-
rate out, and this lowers the melt-
ing point. At 8.5 per cent, of
copper concentration, the solid so-
lutions of silver are saturated, being
incapable of taking up any further
quantity of copper. After that, the
crystals contain varying amounts
of silver imbedded in the melt.
At 40 per cent., the me!t solidifies
about the crystals. The saturation
point for copper is 96 per cent.
Likewise from lOO per cent, copper
to 40 per cent., the crystals vary
, in the amount of copper contained,
p. I At 40 per cent, of concentration,

the solid solutions are in equilib-
rium with the melt, and therefore a eutectic mixture is formed. These
thermal relationships obtained from Guertler's Metallographie are given
in Fig. I.

OPTICAL CONSTANTS OP BINARY ALLOYS.

The optical constants of these alloys are shown in Table I. while Fig. 3
is a graphical representation of the same.

Table I.

Silitr-Copper Series.

Wt-pM-CCBt

Atoai.PatC«ol.

Of Cu.

.202

.7.08

3.44

4.99

.252

11.35

2.86

98.8

10.27

.517

6.51

3.31

84.87

16.49

.492

7.51

3.69

87.68

42.12

.36

6.61

2.40

80.92

62.94

.312

7.57

2.37

82.92

40.00

.244

13.78

3.36

93.26

87.14

.416

7.05

2.93

84.35

93.87

.507

5.71

2.90

80.98

96.99

.643

5.01

3.22

80.26

100

100.00

.620

4.14

2.57

73.11

The concentration-refractivity curve shows a minimum near the eutec-
tic point, the index of refraction being the lowest here excepting that of
pure silver. As the eutectic point
is left in either direction, there is
an increase in the index of refrac-
tion.

The absorptive index-concentra-
tion curve shows a relative maxi-
mum near eutectic point, but the
absorptive index of every alloy is
higher than of copper and always
lower than of silver.

PlaUnum-Silver Alloys.
Similarly platinum and silver
form two series of solid solutions
separated by a gap. This gap con-
sists of a region of heterogeneous
mixture of silver and platinum ex-
tending from approximately 34.8
per cent, to 83.5 per cent, of plat- Fig. 2.

inum concentration. Beyond these

points in either direction, we find solid solutions. These relations are
found in Fig. 2. This was also obtained from Guertler.

l62

LOUIS K. OPPITZ,

'?f. r..

I 9

/•

/••

Fig. 3.

Fig. 4.

The optical constants of these alloys are found in Table II. while their
graphical representation is embodied in Fig. 4.

Table II.

PUUinum-SUver Series,

Wt. Per Ceot.

OfPt.

Atom. P«r C«ot.
of Ft.

Loss io MsM of

Alloy After

Pueiog.

.202

17.08

3.44

94%

0.000 gr.

8.9

.71

5.92

4.26

86.54

0.035

19.18

1.05

3.69

3.91

78.85

0.020

26.97

1.13

2.736

3.09

67.95

0.032

31.18

1.26

2.42

3.10

65.56

0.000

33.83

1.45

2.29

3.33

65.98

0.000

35.64

1.57

2.15

3.39

65.27

0.000

47.47

1.74

1.82

3.18

60.41

0.000

83.39

2.12

1.85

3.95

66.24

0.000

100

100.00

2.03

1.96

3.80

65.61

0.000

Good working surfaces of the platinum-silver alloys were easily ob-
tained.

The concentration refractive index curve shows an unmistakable in-
crease toward pure platinum. The concentration absorptive index curve
indicates a very sudden drop from pure silver to the next member of the
series. After that the decrease is very gradual. The absorptive index

Na"a^'l OPTICAL CONSTANTS OF BINARY ALLOYS. 1 63

of pure platinum is slightly lower than that of the solid solutions of
crystal type II. in Fig, 2.

In general, for platinum-silver alloys as well as silver-copper alloys
when solid solutions are formed, an index of refraction which increases
with the concentration indicates a decreasing absorptive index.

A typical sample of the readings (those on the eutectic alloy of silver
and copper) is included below:

Sample Series of Observations for Mirror No. i

Atomic per cent. Cu ^40. Eutectic Alloy of Silver and Copper.

Angle of Incidence - 74* 7'.

I. Scratches Parallel to Plane of Incidence.

Polarizer at 285** 59'.

28* 10' 47* 20'

25 46 50

10 47 25

20 10

2 20 15

28 17 47 12

Aver. 37* 44'

295* 35' 327** 55'
35 328 00
296 05 327 40
295 40 50
40 45

295 43 327 51

Aver. 311*47'

28* 05' 47* 20'

27 55 05

28 15 15
00 25
15 20

Polarizer at 195*» 59'.

Polarizer at 105** 59'.

Polarizer at 375** 59^.

28 06 47 17

Aver. 37** 41'

296** 05' 328** 05'

295 35 327 40

296 10 328 00

295 35 327 45

296 20 328 15

295 57 328 15

Aver. 311** 57' Aver. 131** 55'

2^ - 217** 47' - 131** 56' - 85** 51'
T - 2^ - 311** 47' - 217** 47' - 94**

204** 45'

230** 30'

205 10

204 50

205 00

204 56

230 39

Aver.

217** 47'

121** 40'

142** 35'

121 23

142 30

Aver.

131** 56'

204** 45'

230** 45'

205 05

204 55

205 10

204 59

230 39

Aver.

217** 49'

121** 25'

142** 15'

121 16

142 34

164

LOUIS K. OPPITZ.

LSntm.

II. Scratches Perpendicular to Plane of Incidence.
Polarizer at 285*» 59'.

28* 10'

46** 50'

47 00

46 55

47 20

28 18

47 06

Aver.

37** 42'

295** 15'

328** 59'

327 40

295 30

327 50

Aver.

31^40'

28** 15'

46** 50'

47 00

2 20

28 22

47 07

Aver.

37** 44'

295** 20'

327** 35'

328 05

327 40

295 33

327 49

Aver.

311*41'

Polarizer at 195* 59^.

Polarizer at 105* 59^.

Polarizer at 375* 59'.

205*00'

230*30'

204 45

205 10

204 55

204 56

230 38

Aver.

217*47'

122* 05'

141* 55'

121 30

142 10

121 24

142 15

Aver.

131*49'

204*40'

230*40'

205 10

204 55

230 42

Aver.

217* 48'

121* 55'

142* 05'

141 50

142 15

121 33

142 13

Aver.

131* 53'

2^ - 217*47' - 131*49' - 85*58'

Compensator Readings,

B«ior« Bxtioctioo.

Aft«r Bxtinctioo.

M«i0 8cal«.

8cal«.

Mai0 8ca]«.

8cal«.

16.00

17.00

15.75

16.00

48
14
09

77
86

Av..

. . . 16.00

17.00

General Average: 16.50, 33 divisions.

no!"^l optical constants op binary alloys. 1 65

Summary of Results.

1. Near the eutectic point, the index of refraction is lower than that
of any other member in the silver-copper series, except that of pure silver,
while the absorptive index is a relative maximum for the same concen-
tration.

2. The indices of refraction of the alloys at the saturation points in
the two regions of solid solutions for silver-copper are higher than that
of the pure metal near these points.

3. From the eutectic point of the silver-copper series, there is a marked
increase in the index of refraction in either direction until saturated solid
solutions are formed. The absorptive index shows a behavior which is
approximately the inverse of that shown by the index of refraction.

4. The reflecting power of a metal of relatively low reflecting power is
in general improved by mixing this metal with one of relatively higher
reflecting power. This is in agreement with the work of others.

5. Whenever solid solutions are formed an increasing index of refraction
indicates a decreasing index of absorption. This is borne out by the
studies of both the silver-copper and platinum-silver series.

In conclusion, I wish to record my grateful appreciation to Dr. H. C.
Richards for placing at my disposal the facilities of the Randal Morgan
Laboratory of Physics. Not only has he shown interest throughout the en-
tire progress of this work but it is to him that I owe my first interest in the
subject of optical constants. It is also with pleasure that I acknowledge
my indebtedness to Dr. E. A. Eckhardt who has aided me with numerous
valuable suggestions in every detail of the work.

The Randal Morgan Laboratory of Physics.
University of Pennsylvania.

1 66 /. A, CILBREATH.

Hi Cl«etroart«f

IONIZATION OF POTASSIUM VAPOR BY ORDINARY

LIGHT.

By J. A. GiLBREATH.

THERE has been some doubt as to whether the vapors of the alkali
metals are photo-electric under the influence of ordinary light
Anderson found that potassium vapor was ionized by ultra-violet light.^
Hughes criticizes his conclusions and suggests that the photo-electric
currents observed by Anderson were not due to the vapor but were surface
effects.* This investigation, suggested by Dr. Anderson, was undertaken
to determine whether or not potassium vapor is ionized by ordinary light.

A tube with parallel plate elec-
trodes of the type shown in Fig.
I was used. The tube was of
ordinary glass impervious to ul-
tra-violet light. The electrodes
were of copper, about 2 cm. wide
by 3 cm. long and separated by
a space of i cm. There was a
platinum cylinder fused into the glass at C and earthed to prevent con-
duction over the glass. The solid potassium distilled in vacuo was in
bulb K. The vacuum at the start was about .005 mm. of mercury.
There was a slit at 5 i mm. wide through which the light was admitted.
The source of light was an arc or a 500-watt nitrogen filled tungsten
which was focused by a lens system on the slit. The currents were
measured by a quadrant electrometer with a sensibility of 4,000 mm.
per volt. When the light and slit were in the position shown in Fig.
2 and so adjusted that the light passed evenly between the electrodes,
very little striking either electrode, distinct photo-electric currents were
observed when a sufficient E.M.F. was applied at B.

With zero potential at B no currents were obtained in any case. When
a positive potential of 317 volts was applied a current represented by
a deflection of i cm. in 25 seconds was obtained. But when the light and
slit were adjusted as in Fig. 3 and the same potential was applied the
deflection was practically the same, namely i cm. in 23 sec. Again when

> Phys. Rbv., Vol. I, No. 3. March, 1913.
« Photo Electricity — Hughes, p. 25.

Na"ai^l IONIZATION OP POTASSIUM VAPOR. 1 6/

the apparatus was adjusted as in Fig. 4 the same deflection was obtained,
namely i cm. in 24 sec. Practically the same currents were obtained in

Fig. 2. Fig. 3. Fig. 4.

the three positions when 396 volts were applied, namely 5 cm. in 5 sec.
At a conservative estimate 50 times as much light fell on the electrode in
Figs. 3 and 4 as in Fig. 2.

The fact that the currents were the same regardless of the quantity
of light that fell upon either electrode would seem to show that the effect
observed could not be due to photo-electric action at either electrode.
The photo-electric current from a metallic surface is proportional to the
intensity of the light falling upon the surface. The presumption is thus
raised that the effect observed is due to the vapor and the electrodes are
not photo-electric for ordinary light.

A second test was made as follows: The bulb containing the potassium
was kept at room temperature and the portion of the tube containing
the electrodes was heated. The result would naturally be to drive vapor
out of the main tube into the bulb. As the rate of diffusion in a tube of
this type and at the temperature used is probably very slow, the temper-
ature was raised to 65® Centigrade and kept there for several hours.
The photo-electric currents were measured at various intervals. The
result was that the photo-electric action at first increased and then
gradually decreased. Neither the tube nor the light were touched nor
altered in any way during this test so conditions were identical throughout
except for the change in temperature and the lapse of time. The effect
of mere lapse of time was tested and found not to influence the activity
of the tube materially. It was found, however, that applying an E.M.F.
and light for any considerable time did reduce the activity. Conse-
quently complete ionization curves were not attempted but two different
E.M.F.s, both positive and negative, were applied and the light was
allowed to shine between the electrodes only for a very brief time during
each test. No perceptible fatigue could result from that amount of use.

The curves / and //, taken for + 280 volts and + 287 volts, and curves
///and / V, for — 280 and — 287 volts, show an increaseand then a gradual
decrease in the photo-electric currents. The ordinates represent the rate
of deflection or the current. The abscissae represent the irregular inter-
vals of time between tests. The temperature of the tube except at the

1 68 J. A, CILSREATH. ^S2

beginning and during the last interval of time was kept at about 65
degrees. At the beginning and throughout the 16-hour interval at the
end of each curve the temperature was about 20 degrees.

These curves seem to show that the effect observed is a vapor effect.
When the temperature is first raised the small amount of potassium which
has collected in the main part of the tube begins to vaporize and the
photo-electric action increases. When all the potassium in the main
tube has been vaporized the density of the vapor there is reduced on
account vapor being driven into the cold bulb. As the density of the
vapor lessens the photo-electric action diminishes.

The effect of changing the pressure without changing the temperature
in the main tube was next tried. It was believed that an electrode effect
would show an increase of current when the pressure decreased while a
vapor effect would show a decrease with decrease of pressure. The bulb
was packed in salt and ice and kept at an average temperature of — 8^
Centigrade for three hours. Brief ionization curves were taken just before
and at the close of this period. Curves V and VII were obtained before
reducing the pressure and curves VI and VIII after. As the photo-
electric effect was plainly reduced it seems that the effect was due to the
vapor.

The reason that much lower voltages were re uired in this last test
than in the previous tests was that the whole tube including the bulb
had been strongly heated for several hours. This greatly increased the
quantity of vapor between the electrodes.

Surface effects were sometimes met with in making these tests. After
the tube had been strongly heated and a narrow beam of light passed
between the electrodes the electrometer showed a negative deflection
with no E.M.F. applied. The beam of light struck the end of the tube
as indicated by the dotted lines in Fig. 5. When the beam was adjusted
to shine between the electrodes without touching the upper or lower wall
of the tube as in Fig. 6 the effect disappeared. It was due to electrons
released from solid potassium at the top or bottom of the tube.

Fig. 5. Fig. 6.

The fact that it is possible to obtain a regular ionization curve for
positive as well as negative E.M.F.s is additional evidence that the effect
is a vapor effect.

* /

(

\ oww

1 XI mn Lout)

•

<•

■/

IX (♦!••

IDIMI

•

M ala

tlOala

•0 aU

Main'

4* aiiT -

ukii-

Fig. 7.

«^

•

i»(-«r

WlM)

iiala.

UOaia

•Oaia

•

■

•1

68im

XZX (-JM

wUt)

•

1 i

f j>

M ala 110 ata Mala

Fig. 8.

I70

J. A. CILBREATH.

.Swntti

In conclusion there are four lines of reasoning which seem to show that
potassium vapor is ionized by ordinary light.

loui (totztzt
Fig. 9.

If]

XX (-••)

Fig. 10.

1. When a beam of visible light is directed first on one electrode then
on the other, then midway between the electrodes the photo-electric
currents are very nearly the same.

2. When the temperature of the main part of the tube is raised and the
vapor driven into the cold bulb the photo-electric currents decrease.

3. When the temperature of the bulb is lowered thereby decreasing the
pressure of the vapor throughout the tube, the photo-electric currents
decrease.

4. The fact that ionization currents of approximately the same mag-
nitude are obtained with both positive and negative E.M.F.s applied,
is an indication that we have here an ionization of the vapor.

Physics Laboratory.

University of Washington.

No!"a. 1 THEORY OP VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS. I 7 1

THEORY OF VARIABLE DYNAMICAL-ELECTRICAL

SYSTEMS.

By H. W. Nichols.

/^^OMPARED with the volume of literature on the subject of electrical
^^ and mechanical systems with invariable elements — resistance,
inertia or inductance, stiffness or capacity — ^very little has been published
concerning systems in which these elements are variable in a general way,
and this notwithstanding the fact that such important applications as
electric signaling depend upon the variability of some element of the
system. The problem is usually to find the effect upon a steady or quasi-
steady state of changes in some element, and the steady state is often
not of interest, so that its existence is ignored. The ignoring of the un-
disturbed state but not of the energy which is transformed from it by
the variable elements leads to many interesting and important problems,
some of which are considered in this paper.

A more formal statement of the problem to be solved is: A dynamical-
electrical system capable of description by means of differential equations
obtained from a Lagrangian and a Dissipation function of the usual type
is operating under the influence of given impressed forces. This state
of the system is disturbed by changes in an inertia, resistance or stiffness
element and the disturbed state is considered by itself, the undisturbed
state being ignored if its motions are of types (defined later) different
from those of the disturbed state.

It will be found that energy can in this way be added to the disturbed
state in a manner similar to that in which it is '' lost " from a dynamical
system by transformation to a type with which the purely dynamical
problem is not concerned, namely the energy of thermal agitation of
systems whose codrdinates are not required to be included in the La-
grangian function in order to obtain a satisfactory solution of the larger
scale dynamical problem.

A very simple illustration of a system of this kind is an electric bell or
" buzzer." From one very practical point of view the dynamical system
of interest comprises only the button with its impressed force and the
vibrating armature, so that the type of motion obtained bears no relation
to that of the source (for example, a battery) nor is there any necessary
relation between the energy of the motion and the work done by the

172 H,W. NICHOLS, [^b£

impressed force at the button. From another point of view the system
is a generator of oscillations, either electrical or mechanical, without
energy supply at the frequency of the oscillations and deriving its energy
from a source which may be ignored if only oscillatory states are of in-
terest.

From the point of view here taken it is convenient to classify dynamical-
electrical systems into the following types, the first two of which are the
ones ordinarily considered and the last two are of particular interest be-
cause such systems are capable of bringing energy into play from sources
of different kinds, as will be explained later.

Types of Systems. — i. All purely electrical systems (whose motions
do not involve the changing of a mechanical co5rdinate) and those in
which the variables are measured from equilibrium positions, the gener-
alized displacements x being small, are characterized by Lagrangian and
Dissipation functions which are homogeneous quadratic functions of the
displacements and velocities and have constant coefficients. As a
result the system of differential equations of motion is a set of linear
equations of the form :

SiiXi — Siipct — *SisXs = ei

— SiiXi + StiXi — 02sXs = et

• •••••

Here any stiffness operator 5 has the form

S = Ipi + rp + s; p ^ d/dt

and « is a given function of the time, being an impressed force. This
system is characterized by the fact that its operational determinant
|'5ii5nn| is symmetrical, since the Lagrangian and Dissipation functions
are of the specified type, and by the fact that it satisfies the energy
principle. Due to these facts, the reciprocal theorem holds, namely,
if C/fc represents the operator which finds the displacement Xj from unit
driving force at the place k, or the mutual compliance between j and k :

where D is the operational determinant of the system and Z>/* the minor
of row j and column A. But when a unit driving force is located at j
it produces at k the symbolic displacement Ckj = Dkj/D; and because
D is symmetrical, Djt = Dkj. Hence the two mutual compliances are
the same. Such a system has been called bilateral.

The set of linear equations above has the further property that, if
the driving forces are periodic and are resolved into their Fourier com-

Nol"^'l '^BEORY OP VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS. 1 73

ponents, all these components will in general appear in the particular
solution for any x, and no others will ever appear. This is because the
coefficients are constant, and this property will be described by saying
that the system cannot change the type of driving force. This feature is
important in many applications.

2. When the co5rdinates of interest are the small departures from zero
values in a state of motion, the Lagrangian function is not a homogeneous
quadratic one but leads to a set of differential equations of the form:^

Snxi - {Bit + Cit)xt - (5is + Cis).r, - • • • = <?i
- (Bn - Cti)xi + Si^i - (Bu - Cu)x^ - . . . = g,,

in which Q* has the form Rfkp = Rkip»

If the kth equation is multiplied by pXk and the results added, the terms
in C cancel and hence do not enter into the equation of activity. Systems
of this kind are different from those of the first class in that the reciprocal
theorem does not hold. They resemble them, however, in reproducing
the type of driving force. While the Cs are of odd order in p, they do
not contribute odd order terms to D, although they do to its minors.
These centrifugal terms correspond formally to mutual resistances, but
differ from ordinary dissipative terms in that they occur in pairs in such a
way as to make the determinant D of even order in p so long as true
resistances are not present. This is suggestive, as it indicates the pos-
sibility of compensating true resistances by similar terms, and the general
conditions under which this may be accomplished will be discussed later.
It is evident that in order to do this there must be a transformation of
energy from that of an ignored type of driving force, otherwise an un-
compensated flow of energy takes place out from the system through the
resistances.

Energy dissipated as heat is of course not lost, but simply transformed
into a type which is ignored in purely dynamical-electrical problems as
outside the scope of the investigation. In the same way energy may be
thought of as entering the system from an ignored source through
suitable devices for changing its type into one with which the problem
is concerned. The principal object of this paper is to investigate systems
in which this occurs.

3. When the differential equations of the system are linear but with
coefficients which are functions of the time, the system is characterized
by the very important fact that it is able to execute motions whose types
are different from those of the driving force, that is, the particular solu-

1 See, for example, Whittaker, Analytical Dsmamics, p. 84.

1 74 ^' ^- NICHOLS. [SSS

tions no longer correspond in component frequencies to the driving forces.
If therefore some driving forces are of types which are ignored for the
purposes of the investigation, perhaps because they have no direct
influence upon certain parts of the system* it is possible to supply energy
to the system in a way similar to that in which it is drawn off as heat.
The definition chosen for the type of driving force or motion is now shown
to be a suitable one, for when no energy is transformed to ignored types
of any kind, the particular solutions depend upon the then pure imaginary
roots of 2? « o.

4. When the differential equations of the system are non-linear, the
principle of superposition no longer holds, which fact is of considerable
importance in some applications. Non-linearity often means that all
the mechanism of the system has not been taken into account.

I. Systems with Invariable Elements.

The classification adopted is a convenient one for our purposes and will
be followed in this treatment, beginning with the first two classes.
These are the cases ordinarily considered, and will be taken up only very
briefly to collect some useful results, all of which, however, are no doubt
old.

The coefficients of />* in the differential equations:

SiiXi — Sii^t — • • • = ^1

— SiiXi + -SmXi — • • • =* f 1

.....

being constants, p may be treated as an algebraic quantity, with the result

that any x has the value

^Dikej

If fy is the only driving force acting.

Dike,'

Xk =

and the operator DjkjD is called the mutual compliance, C/*, between
the parts k and j, being the operator which finds the displacement pro-
duced at A by a driving force at j. The use of such operators apparently
originated with Heaviside. For systems of the first class, Cki = C/*.
When A = J the compliance is self instead of mutual.

In certain special cases in which the driving forces are special functions
of the time, these operators reduce to algebraic quantities. Some of the
important results are considered below.

Na"af'] ^B^ORY OF VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS. 1 75

Steady State. — If in C,*, the operator p is put equal to zero, the solution
obtained is that appropriate to the final steady state under the influence
of a constant driving force.

Harmonic State. — ^When p is put equal to in + 4, the solution is that
appropriate to an exponentially increasing or decreasing harmonic driving
force of frequency n/2T and damping B, and in the extremely important
case in which B ^ o, the solution is that for an undamped harmonic
driving force of frequency n/2T. The highly developed subject of alter-
nating currents and of sustained simply periodic motions in general,
depends largely for its practical value upon this method of reducing the
differential equations to algebraic ones.^

When the driving force is harmonic, putting p = in reduces any x to
the quotient of two determinants whose elements are complex numbers.
Further, since any 5«« enters linearly into the determinant D, any Xk
must be of the form (omitting all subscripts for brevity) :

aS + b

or a bilinear transformation of 5. This is the reason alternating current

loci are circles.

When this transformation is thrown into its three constant form by

division by c (which obviously cannot be zero for any physical system)

it becomes:

a I ad — be A

c c*S + dfc " S + K'

the constants of which should and do have physical significance, namely:

When A = o a change in S has no effect upon x; these two parts of
the system are conjugate. Hence to have conjugate parts a system, if
connected at all, must have at least three degrees of freedom.

/ is the value of x when 5 is infinite, that is, when the branch having
the operator S is removed.

When K is zero or A// reduction of 5 to zero produces infinite and
zero values, respectively, of x.

Many other relations may be found involving this transformation. It
is useful in experimental work, being established by three points.

In practical work, for example in alternating currents, it is usual to
ignore the operational nature of these quantities 5, etc., and treat them
in the same way as the forces, displacements and velocities themselves;
that is, complex numbers, a + ib, are used for both operators (0) and

> For a conciae and valuable treatment of this caae (^ » in + 2) see G. A. Campbell,
Proc. A. I. E. E., April, 191 1.

176 H.W. NICHOLS. [^SSS

physical quantities (T), the latter being functions of the time, which
variable is eliminated in single frequency problems by this method.
Consequently no distinction is made between products such as OT and
TT^ wliich are physically very different things, the first being of frequency
n/2T and the second having a constant part and a part of double fre-
quency, n/x, that is, not being capable of representation in the complex

plane of T. Thus if

0 ^a-^-ib, T'^c + id,

the formal product OT gives a physically intelligible result, while the
formal product TT = CiCt — didi + tCcidj + cadi) has no physical sig-
nificance. In particular, it does not represent power, torque, etc,
Steinmetz avoided this difficulty by giving up the rule * X i = i X ♦
for these products, but it seems better* to introduce a double frequency
operator, say k, represented geometrically by a unit vector at right angles

to I and i, such that

0 = a + ib + ko,

T ^ c + id + kOj

as before, to retain the formal operations Or, but to define the complete
product TT of two physical quantities as the sum of the scalar and
vector products:

Ci Ci

TT+TXT^ CiCt + didt + k

di dt

^P + kQ.

This gives the correct value (in contrast to formal multiplication)
and moreover indicates, by the unit operators i and A, the nature of the
result with respect to frequency. In problems in which both time dif-
ferentiations and multiplications TT are required, there is some advan-
tage in using this method and it will be used here when necessary.

Impulses and Initial Values, — ^When p is made infinite in the compli-
ance operator, the initial values of the coordinates are found.

When a driving force tj is impulsive, its impulse being

/, = Lim i ejdt,

we have e^ = plj, which may be substituted in the differential equations
to find the behavior under this kind of excitation. An advantage of this
method of treatment is that the initial conditions may be found from
the differential equations; thus the initial displacements are

Xk{o) = Lim (pCikli),

pes so

> Armour Engineer, January, 191 2.

No"^*] ^BEORY OF VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS, IJJ

and the initial velocities

In general, since C/* = D^k/D and the order of D is two higher than that
of Djk, the initial displacements are zero, the initial velocities finite, and
the initial accelerations infinite for impulsive excitation. This ceases
to be the case when inertia terms are lacking from some of the elements,
when some initial velocities may be infinite. {pCjk)"^ and if^C,k)~^,
with p infinite, are respectively the initial resistance and initial inertia
offered at the place A to a sudden disturbance acting at the place j.

Free Oscillation. — ^The condition of free oscillation of the system is
that 2^ = 0, which equation gives the values of p ^ in + 6 corresponding
to the frequencies and damping constants of the component oscillations.
If the free oscillations are sustained, 6 = 0.

Since the condition D = ois also the condition that the effect produced
by a given driving force shall be the largest possible, it is clear that the
two requirements of good signaling, namely that the effect x shall be
both a large and a true copy of the cause e for all wave forms, are in
general contradictory; for the condition that x shall be largest is also the
one that the system shall oscillate without regard to the driving force.
In this case the " quality " of reproduction is zero. (It is highly desirable
to develop some dynamical specification of quality of reproduction which
corresponds to and predicts data furnished by the senses.)

There are two obvious exceptions to this statement, one the case in
which the compliance C is of one kind and also independent of p, the
other the case of an infinite number of degrees of freedom, to which this
argument does not necessarily apply.

II. Systems Having Variable Elements.

When the inertia, resistance, or stiffness factors are variable with the
time only, the differential equations of the system are linear with variable
coefiicients and the importance of this class of systems depends, from
our point of view, upon the fact that the particular solutions contain
types of motion different from those of the driving force. The variation
which is most important is that in which the magnitude never departs
greatly from a mean value and in cases of physical interest it is then
possible to find a solution in the form of a convergent series as has been
done, for example, by Barkhausen,^ Pupin and others, using a method
of successive approximation. The object of this paper, however, is
not primarily to find the coefiicients in such an expansion, but to show

1 "Problem der Schwingungserzeugung/* 1907.

1 78 H. W, NICHOLS. [^SS

how the transformation of energy from one type to another leads to
useful results.
Considering any generalized stiffness factor with constant elements,

S^lp' + rp + s,
It is clear that when /, r, and 5 are variable 5 must be written

p{lp) +rp + s^lp' + [r + {pl)]p + s,
so that if

/ « /o + X, r =« fo + P, s ^ So + a,
the change in 5 is

A5 = X/)» + (p + \')p + a.

This interpretation of AS will be understood in what follows.
In the set of linear equations

5iiJCi - Siipct - • • • - ei
- SnXi + 521X1 - • • • » ei

let 5ii be the variable element. This can always be done, if necessary,
by a linear change of variable. Put Sn =» 5© — 55 and call D© the
value of D when 6S = o. Also let

Xk - Xk + ik
in which

If these values are substituted in the set of equations above we get

(5o - d5)fi - 5iifi = 6SX1

— 5iif 1 + 5iif 1 — • . • «s o
" 5iif 1 — 58if 2 + • • • =0

(A)

and any f is therefore:

This equation shows that, so far as first order terms in 6S are con-
cerned, the disturbance superposed upon the unvaried system may be
accounted for by supposing the sources of the unvaried motion removed
and replaced by a driving force equal to 6SX1 whose seat is in the variable

No'af i THEORY OP VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS, 1 79

element, the constants of the system remaining the same. It shows
also that, with a proper interpretation of the symbols, the driving force
for large or small values of 6S is

dSxi

■ - <

since when this is operated upon by Dki/Do the result is (jb*

There is, however, one very important difference between this system
and one without ignored sources and excited by the same driving force,
for in the latter case all the power expended in the system comes from
the driving force, while in the former case the power may come from
impressed forces required to produce the variation 6S (that is, from
the equivalent driving force) or it may come from the sources maintaining
the ignored state, in which case the variable element acts as a transformer
of energy from one type to another. These two r61es are essentially
different ones and will be discussed in detail shortly, after a few con-
sequences of equation (i) are noted.

It follows from that equation that if f is intended to be a copy of the
variation 6S, the copy cannot be perfect unless Dn = o, which means
that the compliance of the system, measured from the variable branch,
shall be zero. Consequently a perfect copy of the variation of a stiffness
factor S cannot be obtained with finite displacements. The terms of
degree higher than the first in 6S indicate distortion, or departure from
perfection of the copy otherwise than through resonant selectivity of the
system. An illustration is a microphone telephone transmitter, in which
the electrical copy of the motion of the diaphragm is desired to be perfect.

It is evident that forces //, of non-ignored type, may be added to the
system in the usual way, and the right hand members of (A) will be
supposed increased by these impressed forces of f-type.

To evaluate f in algebraic terms, 6SX1 must be reduced to a function
of time, say 0(0, and dSDu/Do to the form F{t) • P{p) ; then the solution
is the sum of terms such as

€' - ^«(o, f" = F{t) . p{pn\ r - - F(t)P(pn'\

etc.

A few simple examples will be given to show the application.

(a) Consider an electrical circuit containing inductance /, resistance r,
capacity 1/5 and a constant source of E.M.F. E. The elements are in
series. Let the stiffness of the condenser vary as 5o(i — ^ cos nt). The
steady state is Xi — E/so and

1 8o H. W. NICHOLS. fSSS*

LSbkbs.

(. a cos nt \ aE cos tU

'+—s; J-^;;-'

where

5o = /p» + r/> + So.

Since i/5o operating upon'any periodic function can always be evaluated,
the expansion can be carried out.

(b) If the battery is replaced by an alternator of voltage E cos qt^ we get

Ee*^*
Xi = ^j7T-r ; SSXi = A. cos[(g + n)t + a] + B cos[(g - n)t + p].

Hence f contains terms of frequencies proportional to g db «, and higher
order terms of frequencies proportional to q zk kn, k = i, 2, • • •.

(c) If, in this circuit, the resistance is varied according to r(i — acos«/)»
we have 6SX1 = ar cos nt • pXi and no disturbance is produced unless
Xi is a function of the time, that is, unless the E.M.F. £ is variable.
If, however, the condenser is shunted by an infinite perfect inductance
we have, for a constant E.M.F.

Xi = E/rp; 6SX1 = a£ cos nt.

(d) If, in the last circuit, the inductance is variable so that

6S == al cos nt * p ^ anl sin nt • p
with Xi = Ejrp we get

bSXi =» — aEnl/r • sin nt,

and the part al cos nt • p of the stiffness dS has no influence because the

current pXi is constant. Finally, if the circuit carries an alternating

current

pXi = A cos qt,
we find

dSXi = B[(q + n) sin {q + n)t + (g - n) sin {q - n)/].

So far no account has been taken of the manner in which the variation
bS is produced, while if a complete description of the behavior of the
system is to be had, the dynamics of the variable element must be
included in the system of equations. If the energy represented by the
f-system comes from forces required to produce the variation 55, the
principle of energy will be satisfied by including these forces and no
liberation of energy from the original state will take place. If, on the
other hand, the energy of the f-system comes from the ignored state and
is simply set free by the action of forces producing the variation, the
principle of energy will not be satisfied for that system and there will be
no particular relation between the energy of the latter forces and the

No'a^] THEORY OP VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS. l8l

energy set free by them. This is a very important distinction; for ex-

ample» in the problem of the telephone amplifier and of generators of

sustained oscillations the energy which is transformed must come from

an auxiliary and ignored source.

To determine the source of the energy of the disturbed state, consider

how variations in any element are produced. Any inertia, resistance or

stiffness factor, or in the electrical case, any inductance, resistance or

capacity, is fixed by geometrical co5rdinates and by quantities of the

nature of permeability, dielectric constant, etc., depending upon the

properties of materials. The geometrical co5rdinates and electrical

charges are the variables chosen to represent the state of the system,

together with these material constants whose dynamical natures are

either not known or supposed not known. With these the Lagrangian

and Dissipation functions are built up, the equations of motion being

then found from

d dL dL SF ^

di dx dx dx

Now L is a function of the coordinates x and their velocities x and is
differentiated by each to find the reactions; hence any change in the
geometrical shape of any system of electrical conductors or other bodies,
and the forces thereby brought into play, are included in the dynamics
of the system. It therefore follows that any energy derived from the
change of geometrical form of any inductance, capacity, inertia or stiff-
ness of the system comes from impressed forces tending to change this
geometrical form and is taken account of in the equation of energy. In
particular, in any complete cycle of operations, no energy comes from
the undisturbed state of the system, hence no energy of an ignored type is
continuously transformed by the variation of the geometrical codrdina^s
determining any inductance^ capacity, inertia or stiffness factor of the system.

On the other hand, forces due to the variation of material constants in
the Lagrangian function L are not included in the dynamical equations
obtained by Lagrange's method and hence the energy set free or trans-
formed by their variations may come from that of the undisturbed state.

An example of this is found in the case of a deformable inductance coil.
If the coil is energized by a battery and then deformed so as to vary
periodically the inductance of the circuit and thus produce an alternating
current in it, all the power represented by. that alternating current will
be derived from mechanical forces required to vary the shape of the cir-
cuit. Such a device could not be used to bring into play an auxiliary
source of energy of different type — for example, it could not be made
into a telephone amplifier. An ordinary telephone receiver is also a
system of this kind.

1 82 H.W. NICHOLS, [^SS

The same thing is true in the case of a condenser whose geometrical
dimensions only are varied, for here the forces resisting deformation are
derived from the Lagrangian function and enter into the activity equation.

If in the coil the permeability of the medium is varied without mechan-
ical motion of it as a whole, and hence the inductance varied without
varying any geometrical codrdinate entering into the Lagrangian func-
tion, the energy of the varied state must be derived from the battery.
It may require energy to produce the variation in permeability, but the
amount of this energy may be quite different from that transformed from
the auxiliary source and will depend upon different things. The same
remarks apply to the variation of the dielectric constant of a condenser
without bodily motion of the medium itself.

The Dissipation function F is also a function of the co5rdinates and
their velocities, together with constants of materials, but in the equations
of motion only its partial derivatives with respect to the velocities appear.
Hence if either the geometrical dimensions of resistances or the specific
resistance constants of materials vary with the time, the impressed forces
required are not part of the dynamical scheme described by the Lagran-
gian equations. The energy of the disturbed state must come from the
sources of the undisturbed state, while the energy required to vary the
resistances need have no necessary relation to that brought into the
system from the ignored sources. Any resistance variation, however
produced, is able to transform energy from the undisturbed state to the
disturbed one.

From this discussion it follows that the only variations not already
taken account of in the ordinary equations of motion are those in which a
permeability, density, dielectric constant, elastic constant, or a resistance
is changed. These are therefore the ways in which energy can be trans-
formed from the sources of the undisturbed state, and in what immedi-
ately follows the variation bS of the stiffness will be supposed to contain
explicitly only these parts, any other part being already included in the
equations of motion as obtained by Lagrange's method. These vari-
ations, as well as the resulting motions, will first be supposed small in
order that the equations may remain linear.

To take account of the variations produced in the way just discussed,
we will now suppose that bS depends upon some mechanical or electrical
co5rdinates X\"' x^, x^^i • •. • Xjf+^ and consequently upon f i • • • ^j^^^
in a way described by the differential equation

which is apparently sufficiently general to include all cases in which the
equations remain linear.

No*^*] THEORY OP VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS. T83

The variation 6S may depend upon M coordinates not required for the
specification of the original system which is not concerned with the
mechanism producing this variation, as well as upon the N co5rdinates
originally required. (We might also suppose that a number of elements
were varied, with

but it follows from the previous discussion that no generality is added
thereby.)

The system (fj^+i • • • fjyr+jr)» which represents the additional mechan-
ism by which 6S is determined, will be supposed subject to laws capable
of statement by linear differential equations, and the driving force
6SX1 will be written

The differential equations of the system, including impressed forces
of f-type, will now be of the form:

(5n - QnHi - {Sit + Oi«)f« {Si^+QinHk

— Oi.-y+ifiyr+i-"=/i

— Sti^i + Sn^i 52, jf^if

0 • • • =/j

0 0 • • • 0

• • . • . •

. . • •

0 0 ... 0

Aside from the fact of the variability of Sn (which variation may be
supposed to be small) this system .differs from those discussed in part I
in that there is no necessary relation between 5u + Qiu and 5*i. Con-
sequently the equation of activity is not satisfied for the ^system, but energy
is continuously drawn into the system from the ignored sources through the
periodically variable element. The energy so drawn is, per unit time:

/*9*

We have now succeeded in making 5*/ different from Sjk in a way not
accounted for by centrifugal terms and such that Sju — Skj may have
practically any form, depending upon the dynamical nature of the
mechanism by which S is varied. This system obviously need have no
special relation to the original one; in fact, if no impressed forces act
upon the additional co5rdinates ^^^\ • • • f j^+j^ required to completely
determine 55, that system has no effect upon the solution for any original
codrdinates except through the Q-terms, for if

1 84 B. W. NICHOLS. [SS2?

any original codrdinate is

f * ^ pCAO • * = I • • • -^.

and contains no explicit reference to the additional mechanism producing
the variation bS.

Since we are treating the original system {1 * * - (jr as the one whose
dynamical nature is known, or as the only one upon which measurements
are to be made, it is appropriate to eliminate the codrdinates {j^+i • • • is-^-M
from explicit appearance; moreover this is suggested by the form of the
determinant above. For these reasons imagine the last M equations to
be solved for the codrdinates appearing in them. Let the result of this
be substituted in the first N equations and put QN+iis'¥i "^ gih the ^'s
being expressed in this way in terms of the impressed forces /jy+i* • -Js-^m'
We then get for the equations of motion

(B)

— 5iif 1 + Snii • • • = /t

" Ssiii "' • • • + Sjfjf^jf = fff.

which contain explicitly only the original variables, and from which the
following fundamental theorem is obtained :

The effect of the ignored sources and of the mechanism by which the small
varicUion bS is produced is to make the original system appear from dy-
namical-electrical measurements on the ^-variables only as one in which Sn
is changed to Sn — Q\u any Sik to 5u + Qik, and the impressed force /i
iofi-\- gi.

These equations are the fundamental ones in the study of systems to
which energy is added by transformation from ignored sources, and to
make clear the meaning of the terms a few special cases will now b^ con-
sidered. First supposed that no impressed forces act upon the codrdinates
is-^i • • ' is+M so that gi = o ultimately. Also suppose all the ^'s are
zero except ^n which will be written af^ + bp -{• c. The equations then
show that the system behaves as one in which the stiffness 5ii is
{III — o)t^ + {fii — b)p + {s\i — c) so that the inertia, resistance and
stiffness, or the inductance, resistance and (capacity)"^ have been de-
creased by the (positive or negative) amounts a, 6, c, respectively. The
quantity —6b called a '* negative resistance " for obvious reasons.
The power added to the system by means of it, or the negative power

NS"af*] THEORY OP VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS. lS$

dissipated by it, is &(i', hence proportional to the square of the velocity
|i as is the heat produced in the resistance fn. Energy is thus brought
into the system by the same general type of ignored mechanism as that
by which it passes out of the purely dynamical or electrical system.

The effects of a and c are changes in the stor^^e of power and hence in
the resonant frequencies and phases of forced oscillations. An ordinary
electric bell is a simple example of this kind of system, especially if the
contact is shunted with a resistance of a few ohms so that the resistance
changes are not too large.

Second, if every Q is zero, but gi is different from zero we have the
simplest case, namely, energy added through the trigger or relay action
of impressed forces which vary only the eliminated codrdinates and do
not act upon the original system. A variable resistance telephone trans-
mitter and an electric switth are illustrations. The power added is gi|i.

Third, if every Q is zero except one with unlike subscripts say Qn,

and if gi = o, the most noteworthy effect is that the determinant of the

coefficients is neither symmetrical nor has it only the special skew-

synunetric elements appropriate to centrifugal forces. The reciprocal

theorem does not hold and we get, for example, putting A for the value

of D when Qit = o:

Cti Ati — QitMt

Cit All

where if s is a second minor of A. The mutual compliances may there-
fore be made widely different in two opposite directions through the
S3^tem. Similar expressions obtain for other mutual compliances.
Examples of systems having these characteristics will be worked out
later in this paper.

Free Oscillations, — One of the most important cases is that in which
all the impressed forces are zero. In this case no forces of f-type act
upon the system and if it is to move and do work all the energy required
must be transformed from the ignored sources. Such a system is called
an oscillation generator.

If the co5rdinates are not to be zero the determinant D must vanish and
its vanishing will determine values p =^ pu p2t etc., which give the
frequencies and damping constants of the oscillations. Now for the
invariable systems occurring in pure dynamics it can be proved that the
real parts of the roots of Z> = 'o are negative if there is any resistance in
the system, so that sustained free oscillations of those systems cannot
take place. This proof, however, does not apply to systems having the
more general determinant here found, and we may expect sustained
oscillations under proper conditions.

1 86 H. W. NICHOLS. [&S^

The condition of sustained free oscillation is that Z> = o with p = in.
Suppose p is given this value in the equation Z> = o which will then
become an equation in n with both real and imaginary coefficients. This
equation is equivalent to two equations, say

F{n) = o. G{n) « o.

which when solved simultaneously will give certain values of n and certain
corresponding relations between the n's and the Q*s, The latter relations
are those which must exist in order that p shall be pure imaginary, or
that the oscillations of frequency n/2T shall be sustained. Hence in
order to make the system perform certain free oscillations n, the trans-
forming mechanism must be adjusted to give the corresponding values
of the Q% and by changing these values different oscillations will in gen-
eral be possible. In this respect this kind of system differs from one
merely devoid of resistances, which oscillates simultaneously in all possible
modes except in special cases of normal codrdinates, when special starting
conditions are required.
The power dissipated in a system executing harmonic oscillations is

and the power transformed is

In the sustained free oscillations of the system these are equal, hence
in that case the variable element transforms just enough power to supply
the dissipation.

Since the effect of the variable element is to supply the dissipated
power, it might be thought that to calculate the frequencies of sustained
oscillation it wotild be necessary only to ignore all resistances, or better
the dissipation in each branch by making the resistances zero or infinite.
But the frequencies so obtained will not in general be the correct ones,
even when the variable element introduces only negative resistances
and does not change any reactance. As an example consider the case
of a transformer:

Lip^ + Rip + Si-bp - Mp^

- Mp^ Ltt^ + Rtp + 52

We get, with all losses suppressed :

(LiL, - M^)p' + {SJ.I + SiU)p^ + 5i5, = o,

while equating terms in odd and even powers of p separately to zero

gives the additional term in the frequency equation :

R^{Ri - b)p'.

D =

= o.

Na"a. ] THEORY OP VARIABLE DYNAMICALr-ELECTRICAL SYSTEMS, 1 87

'The odd order terms are

and these equations are inconsistent with Rt(Ri — 6) =» o unless Jlf = o.
Moreover let ± Pi and d= />j be the roots of the even part of 2> = o
and substitute them in the equation for 6. The result will be that in
general i(/>i) will be different from 6(/>j), so that the system, with a given
value of bf will perform only a part of the possible oscillations.

— sAAAAAq
rWAAAAr

Fig. 1.

Example, — ^A device which illus-
trates this theory and method is
one for producing small altemat- i °

ing currents for laboratory use
and known as the '' microphone
hummer." It consists of a bar
B supported on knife edges above
a magnet and carrying a carbon
cell C through which flows current from a battery. Motion of the bar
varies the resistance of the cell and consequently introduces an E.M.F.
into its circuit which produces current in the magnet and sustains the
oscillations under proper conditions. SufHcient energy is transformed
from the battery to allow alternating current to be drawn off into a
load resistance R, To solve this problem, take the case of the slightly
more general arrangement shown in the next figure in which Sw represents
any kind of coupling of the meshes i and 2 carrying the mesh currents
{i and \t. A' and A" are infinite perfect inductances to restrict the
direct currents to theirproper paths. Impressed forces are included
as shown.

This system has a variable inductance in the magnetic circuit since the
permeance of the magnetic circuit is a function of the displacement (|.

Since this variation is due to
a change in the geometry of
the figure it will be taken ac-
count of in the Lagrangian
function as will be seen below.
The inductance of the mag-
netic circuit is I»o/(i — a{i)
where Lo is the average in-
ductance, hence that part of

^A^

I 3L^>-l

Fig. 2.

the kinetic energy which depends upon the magnet is

Ji(/, + I,)* - }io(i + af,)(/, + !,)»,

1 88

H. W. NICHOLS.

where /j is the steady current through it. We therefore get, to fii^t
order terms:

aLiJtp^t + constant = 0^f j + constant

dT
d dT

and the equations of motion are therefore:

5iifi - Snit o - dSXi +/i = Oufi +/i

Here 55X1 is equal to the product of the direct current, /i, through
the microphone and its resistance change 6r.
The determinant of the system is

D ^ — 5ji 52J 0^

o — 0/> Sii

It shows how the centrifugal terms <t>P appear due to the fact that the
pull of the magnet is proportional to the square of the total current, and
also how the symmetry fails when power is added through the term

Since the general theory is inmiediately applicable to the problem,

only some very simple cases will be treated further. Suppose, for

example, that the load is pure resistance, 5i "^ Rp^ and that the bar has

effective mass m and stiffness 5, its resistance being neglected. Also

let the coupling Sw — 5si be through a transformer of self inductances

/, /, and mutual inductance M, The equations of the system now

become :

Jp' + Rp -Mp' -Q fi

- Mp^ K^ ^p /,;

o - 0/> m^ + s ft

K ^ L + J.

Free Oscillalions, — For this case, Z> = o with p = in, giving two
equations with even and odd powers of p. Put Q ^ A + Bp where
A and B are even functions of />; then we get for Z> = o:

G(mp^ + s) + Jil^ = M<t>B,

RKimp' + 5) + i2«* = Mit>A,
Here.! = ao + atp^ + • • • , S = io + 62/^* +

JK" AP.

Na"af*] THEORY OP VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS. 1 89

These equations give a great deal of information; for example we may
inquire what are the conditions under which the bar will oscillate in its
own natural frequency, for which — /)* = sjm = n^. For this case
A = Ril>/M, B = Jil>/M, and we get for the simplest dynamical con-
nection between the resistance change and the codrdinate (s:

6SX1 = 6rli = Of I = (^/M: + pJ<t>IM)ii.
Hence the resistance change must depend upon the displacement and the
velocity according to the law

^^ ' Mil dt ^ Mil ^'

in order that the oscillations shall be sustained and have the required
frequency. In the instrument the microphone is mounted upon an arm
which can be set at various angles to the axis of the bar and this allows
the correct adjustment to be approximated. The resistance change
probably depends upon the acceleration also, which, for this motion, is
proportional to the displacement. It is easy to see that the power drawn
into the system from the battery is equal to that dissipated in the load -R.
We might also wish to know the frequency at which the system could
oscillate for a given dynamical connection; as an example suppose the
microphone is so fastened to the bar that the resistance change is pro-
portional to the displacement. Then A = a©, -B = o and

(-f)-

The frequency is therefore increased and the current /i must be adjusted
to give ao the proper value as set by the second equation. Note that if
0 = 0, that is, if the bar does not react upon the magnet, the frequency
will be «o/2t and no power will be required from the battery to maintain
the oscillation. If the damping of the bar is not assumed to be zero
more interesting problems arise which may easily be worked out but are
too long to be included here. In that case the damping of the bar has
considerable influence upon the frequency (not the same as in the
damped free oscillation of the bar alone) and this effect can be noticed
in the instrument by damping the bar without adding to its inertia.

Forced Vibrations. — Imagine an alternator of frequency «/2t acting
in the mesh i. The current in the load R will be

. pDiif pAiif

D A - QAiz '
where A is the value of D when Q ^ o.

190 H, W. NICHOLS, [

Now D/pDn is the impedance offered by the system when measured
from the terminals of the alternator, hence the effect of adding the trans-
forming device is to change this impedance from

pAii

The impedance may therefore be considerably decreased and also given
very different reactance characteristics. This fact may be described
by saying that the transforming device introduces into one of the ter-
minals the negative impedance — QZ(Ais/A). In fact, this is what would
be indicated by measurements made with a Wheatstone bridge at those
terminals.

If Q(AufA) « I we have Z> = o and the system offers no impedance
at all. This is the case of free oscillations again.

Similar results obviously will be found if the alternator is connected
into the other mesh or if a mechanical force acts upon the bar. The
effect of the transforming mechanism is therefore to amplify the effect
of an impressed force by supplying energy from a source of another type
which has been ignored in the problem.

We may look at this problem in another way: thus suppose only elec-
trical quantities can be measured so that the system is taken to be one
of apparently but two degrees of freedom, both electrical, and information
is to be gained only by operations upon them. Elimination of the me-
chanical coordinate by means of the last equation gives:

(Jp^ + i?/>)fi - (up' + ^^^) f« = /i + QUKmp^ + sh

This system, especially in the neighborhood of mp^ H- 5 = o, would act
very differently from a purely electrical system. The two mutual com-
pliances would be widely different (but still only if Q were not zero),
the effect of the force/ would be changed under the same circumstances,
and for any finite coupling 0 the effective stiffness factors would be
changed.

Another Example. — ^A good illustration of free vibrations occurs in
the case of the " howling telephone " which is formed by holding an
electrically connected telephone transmitter and receiver together as
shown in Fig. 3. the geometry of the system being there made as simple

-l

NaTa"!^! THEORY OF VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS. I9I

as possible. The fact that the dynamical connection between the re-
ceiver diaphragm and the variable element is a column of air (assumed
to move in one dimension only) adds interest to the problem.

Let the variable current be |i and the
displacement of the diaphragms (s and
f I. Then 6r will be A{s, say, and \_f

so that Qjhl expresses the dynamical
connection between the two diaphragms.

We may suppose A to be a mmiber, p. 3^

although this is not necessary. The
equations of motion are now similar to those of the last example, namely

{Lt^ + Rp)^i -{Q- 4>pnt = o.

- ^f I + Stt^t « o,

in which Su means the stiffness offered by the diaphragm to an im-
pressed force and consequently includes the loading due to the air column.
The condition of free vibration is

(Lf^ + Rp)Stt = <t>p{Q - «/>)

or for sustained oscillations

{inL + R)S^ = <t>(Q - tn^),

and to solve the problem we must know Sit and Q for harmonic motions.
These depend ohly upon the air column with the two diaphragms and
may be found in two ways. To do this directly from the differential
equations of the fluid motion, consider the tube of air and let 5o and Sq
be the stiffnesses of receiver and transmitter diaphragms alone, and
also put:

p = mean density of fluid,
6P = increase of pressure over mean,

V = velocity potential in fluid,
qV^dVjdx,

f = impressed force on unit area of the receiver diaphragm.
The differential equations of the fluid motion are, for p = in

(n« + aV)7 = o; bP ^ - inpV,
from which we get

F = e^*(^cos^+5sin^),
«P= -tnpe^*(^cos^+3sin^)

192 H.W, NICHOLS.

The boundary conditions are:
At jc = o 5of 1 ^f - 6P,

Pit = qV.
At :r = / 5«f 8 = «P,

Pit = SF.

These four equations allow us to eliminate A and B and solve for (s
and (s by means of :

(fU fil\ til

5ocos— - anp sin— jfi + 5gfi -/cos — ,

(f^ fil\ , fil

So sin — h anp cos — I f j — anp f s =» / sin — .
a a/ a

From these we get:

(5o5, - ahiY) sin — + anp(5o + 5«) cos —
5„ =

anp cos — + o fl sm —

hi « anp

^ "■ n/ n/'

anp cos — h -4 « sin —
'^ a ^ a

and the problem is solved provided the resulting transcendental equa-
tions can be solved for the values of n and hL

These values of 5m and Q are interesting in themselves: thus 5m « o
gives the free vibrations of a pipe with arbitrary terminal conditions,
and by putting, in the equivalent expression,

nl anp(So + 5«)
tan— =»

a {anpY — 5o5 « *

zero and infinite values of 5©, 5^, we get the frequencies of open and
closed pipes. Other values give the effects of yielding ends.

There is one value of diaphragm stiffness which is unique and important
Suppose Sq = ianp so that the transmitter diaphragm offers only the
resistance ap =« ro. Then

5j2 = 5o + tnro,

tan- =...

and we get the result that no finite free oscillations exist, so that no re-
flection takes place at 5^, all the energy sent out is absorbed, and the

No'a^*] THEORY OP VARIABLE DYNAMICAL-ELECTRICAL SYSTEMS. 1 93

tube behaves as one of infinite length. We therefore get the effect of an
infinite column of fluid upon, a vibrating diaphragm and the energy
radiated to infinity. Hence fo may be called the radiation resistance of
the fluid. For air it is about 40 C.G.S.

Although the equations of free vibration cannot be solved in general,
the character of the solution can be seen. There will be a number of
values of n which satisfy the frequency equation, and for each value of
n a corresponding value of hi which is required to sustain the oscillations
of that frequency.

These examples are sufficient to show the applications which may be
made of this theory and method. Others, together with a treatment of
non-linear connections, will be given in another paper.

194 TH^ AMERICAN PHYSICAL SOCIETY. \sSml

PROCEEDINGS

OF THE

American Physical Society.

Radiation and Atomic Structure.^

By R. a. MnxiKAN.

WHILE the study of the physical and chemical properties of matter has
produced our present atomic theory and furnished most of the in-
formation which IS available about the way in which the myriad molecular
structures are built up out of their atomic constituents, it has been chiefly
the facts of radiation which have provided reliable information about the
inner structure of the atom itself. Indeed, during all the years in which the
dogma of the indestructible and indivisible atom was upon the stage, it was
the complexity of the spectra even of simple gases which kept the physicist
in the path of truth, and caused him continually to insist that the atom could
not be an ultimate thing, but rather that it must have a structure and a very
intricate one at that — as intricate, in Rowland's phrase, as a grand piano.

Yet the evidence of spectroscopy, though tremendously suggestive in the
series relationships brought to light between the frequencies of the different
lines of a given substance, was after all most disappointing in that it remained
wholly uninterpreted in terms of any mechanical model. No vibrating system
was known which could produce frequencies related in the manner correspond-
ing to the frequencies found even in the simplest of series, viz., the Balmer
series of hydrogen. The discovery and study in the late nineties of corpuscular
radiations of the alpha and beta type, with the changes in chemical properties
accompanying them, merely served to confirm the century-old evidence of the
spectroscope as to the fact of the complexity of the atom, and to educate the
public into a readiness to accept it, without at first adding much information
as to its nature. These studies did reveal, however, two types of bodies, the
alpha and beta particles, as atomic constituents, though they said nothing at
first as to their number, their arrangement, or their condition within the atom.

It was the study by Barkla of a radiation problem, namely the problem of the
secondary X-radiations scattered by atoms, which furnished the first important
evidence as to the number of electronic constituents within an atom. He found

> Presidential address delivered at the New York meeting of the Physical Society. Decem-
ber 27, 1916.

No'a^'] ^^^ AMERICAN PHYSICAL SOCIETY, 1 95

that the number of electrons which can act as scattering centers for X-rays is
about half the atomic weight.^ This conclusion was brilliantly confirmed by
the simultaneous study in the Manchester laboratory of the scattering of the
alpha rays in passing through matter,* and out of the converging evidence of
these two types of research there emerged with considerable definiteness the
Rutherford nucleus atom, consisting of a central, positively charged body of
extraordinarily minute dimensions, its diameter being not over a ten thousandth
of the diameter of the atom, surrounded, in the outer regions of the latter, by a
number of negative electrons equal to about half the atomic weight. In this
statement ** the diameter of the nucleus " means the diameter of that portion
of the atom which is found by experiment to be impenetrable to the alpha rays,
while the diameter of the atom means the average distance of approach of the
centers of two atoms in thermal encounters.

But it was again the study of a radiation problem which had to be called
upon to furnish unquestionable information as to the exact value of this
number, and at the same time to provide the most convincing evidence that
we have of the general correctness of the. conception of the nucleus atom.

In a research* which is destined to rank as one of the dozen most brilliant in
conception, skillful in execution, and illuminating in results in the history of
science, a young man but twenty-six years old threw open the windows throujg;h
which we can now glimpse the sub-atomic world with a definiteness and cer-
tainty never even dreamed of before. Had the European war had no other
result than the snuffing out of this young life, that alone would make it one
of the most hideous and most irreparable crimes in history.

For the proof that there exist but 92 elements, from the lightest known one,
hydrogen, to the heaviest known one, uranium, and that these are built up
one from the other by the successive addition of one and the same electrical
element to the nucleus, this proof comes alone from Moseley's discovery
(checked and extended as it has been by de Broglie clear up to uranium) that
the square roots of the characteristic X-ray frequencies of the elements progress
by almost exactly equal steps from the lightest observable one to the heaviest.
Moseley proved this in a general way for both the alpha and the beta emission
lines of the hardest characteristic X-rays of the elements, the so-called K-rays,
and also for the alpha and beta lines of the next softest series, the L series.
Fig. I shows the now familiar regular progression of the wave-lengths of both
the K and the L lines, as the elements which produce them rise in atomic
weight and atomic number from top to bottom of the figure. These photo-
graphs, which are due to Siegbahn, are introduced merely to pave the way for
the discussion and to enable comparison with Fig. 2, which represents the
absorption spectra in various substances of the radiation emitted by tungsten.
These beautiful photographs have just been sent me by de Broglie.* They are

> Barkla, Phil. Mag., 21, 648, May, 191 1.

* Rutherford, Phil. Mag., 21, 669, May, 191 1.

* Moseley, Phil. Mag., 26, 1024, Dec., I9i3> and 27, 703, 1914*

* See Compt Rendu, 165, 87, 352, 191 ?•

196 THE AMERICAN PHYSICAL SOCIETY. [ISS

taken by de Broglie's own method of the continuous rotation of the crystal
of the X-ray spectrometer.

They show the general radiation of tungsten as a background in all the photo-
graphs, and the two K lines of tungsten (W) also in all, while the L series of

Ag.
tungsten appears in the upper one. The edge of the band marked rr- is the

exact point at which, with increasing frequency, the general radiation consti-
tuting the background begins to be absorbed by the silver atoms (atomic
number 47) in the photographic plate, and the point for all frequencies above
which it is absorbed in the remarkable manner discovered by Barkla to be

Br
characteristic of the absorption of X-rays. The edge marked 77- is the corre-

sponding point for the bromine of the photographic plate whose atomic number
is 35. These two points, characteristic of the AgBr emulsion of the plate,
appear on most of the photographs. Absorption of course appears in the
photographic plate as a lightening of the background, elsewhere as a darkening.
The way in which this outer edge of the absorption band moves toward the
central image as the atomic number increases in the steps Br 35, Mo 42, Ag 47,
Cd 48, Sb 51, Ba 56, W 74, Hg 80, is very beautifully shown, in de Broglie's pho-
to^aphs, clear up to mercury, where the absorption edge is somewhat inside
the shortest of the characteristic K radiations of tungsten. This latter line
coincides, nearly if not exactly as will be shown in Table I., with the absorption
edge of tungsten. There must be 12 more of these edges between mercury and
uranium, and de Broglie has measured them clear up to thorium, thus extending
the K series from N => 60 to N = 90, an enormous advance. The absorption
edges become, however, very difficult to locate in the K region of frequencies
because of their extreme closeness to the central image. But Fig. 3 shows
the L-ray absorption bands in uranium and thorium. Fig. 4 shows how closely
these absorption edges follow the Moseley law of equal steps for as many as
twenty steps. In going from bromine, atomic number 35, to uranium, atomic
number 92, the length of the Moseley step does change however by a
few per cent. The data given in Table I. bring out the exceedingly inter-
esting relation that the absorption edge coincides exceedingly closely
in every case with the shortest emission K line of the absorbing substance,
while in the L series one of the two absorption edges coincides exactly in every
case with the shortest emission beta ray of the L series. The other coincides
also in every case with an emission line, though the data is yet too meagre to
permit of a generalization in the case of this second L absorption band.

Now it IS these radiating and absorbing properties of atoms and these alone
which justify a series of atomic numbers differing from and more fundamental
than the series of atomic weights. Our present series of atomic numbers is
simply this Moseley series of steps based on square root frequencies. It is true
that a series of atomic numbers coinciding with the series of atomic weights
was suggested earlier, indeed 100 years earlier, by Prout, and by many others
since then, and it is true, too, that changes in the chemical properties of radio-

No. 3. J

THE AMERICAN PHYSICAL SOCIETY.

197

active substances accompanying the loss of alpha and beta particles led van
den Broek,^ just before Moseley's work appeared, to suggest that position in
the periodic table might be a more fundamental property than atomic weight,*

Table I.

Comparison of Ka and Kb*

BteoMnt.

ICa^

Kfi,

Blcm«iit.

KA'

Kfi,

.914

.367

(.380)

.810

.813

.338

(.345)

.764

.767

.325

(.333)

.681

(.695)

.310

(.319)

.645

.657

.298

.(304)

.611

(.620)

.150

.503

.147

.479

.488

.143

.458

(.466)

.139

.419

(.419)

.135

.399

.408

.130

.383

(.396)

.098about

Comparison of La and Lfi.

Bl«ai«iit.

LA'

•

AT.

Bl«in«nt.

LA'

Lfi.

78
79
82

Au
Pb

1.067

1.037

.945

1.072

1.035

.948

90
92

.756
.719

.750
.702

but since this position is in some instances uncertain, and since the number of
elements was wholly unknown, no definite numbers were or could be assigned
to all the elements until Moseley's discovery was made, and the only evidence
which we now have as to just how many elements there are between hydrogen
and uranium, and as to just where each one belongs, is the evidence of the
X-ray specta. It is true that between helium, atomic number 2, and sodium,
atomic number 1 1, we have no evidence other than the order of atomic weights,
the progression of chemical properties, and the number of known elements in
this region, to guide us in completing the table, but since in the region of low
atomic weights the progression in the Moseley table is always in agreement
with the progression in the periodic table there can be little doubt of the correct
number of each element even in this region which is as yet inaccessible to X-ray
measurements. Moseley's name must then be set over against one of the
most epoch making of the world's great discoveries. And I wish to call atten-
tion to some important conclusions as to atomic structure which are rendered
extremely probable by it.

The first is this. If we may assume that the ordinary law of inverse squares
holds for the forces exerted by the atomic nucleus on negative electrons near

* Van den Broek. Phys. Zeil., XIV.. 32, 1913.

* ThU however was not a new suggestion, see. for example. A. J. Hopkins, J. Am. Ch. S.,
1027. 1911*

198 THE AMERICAN PHYSICAL SOCIETY, [ISS

it — and this time-honored law, so amply verified in celestial regions, has been
fully verified for sub-atomic regions as well by the work done at the University
of Manchester on the scattering of alpha rays — then the Moseley law that the
square roots of the highest frequencies obtainable from different atoms are
proportional to the nuclear charge^ means, without any quantum theory*
that the distances from the nucleus of each type of atom to the orbit of the
inmost electron, if there be such an orbit, is inversely proportional to the charge
on the nucleus, i. e,, to the atomic number. To see this it is only necessary
to apply the Newtonian law connecting central force «£, orbital frequency n
and radius a, namely

— -(2T«)««a. or ---g^ (I)

and then to set as the statement of Moseley *s experiment

^n, Ex ^

when there results at once from (i) and (2)

It may be objected that in the setting up of these relations I have made
two assumptions, the one that the electrons rotate in circular orbits, and
the other that the observed highest frequencies are proportional to the
highest orbital frequencies. The first assumption is justified (a) by the fact
that the recognized and tested principles of physics give us no other known
way of providing a stable system, (6) by the experimental facts of light (Zeeman
effect) and (c) by the phenomena of magnetism, especially the recent ones
brought to light by Einstein and de Haas,* and by Barnett,' which well-nigh
demonstrate the existence of permanent and therefore non-radiating electronic
orbits. The exact circular form for the orbit is a secondary matter upon which,
as will appear later, it is not necessary to insist. The second assumption, that
the frequencies of the corresponding emission lines in the spectra of the various
atoms are proportional to the orbital frequencies, is from a priori considerations
probable and from certain theoretical considerations to be presented later
necessary.

A second conclusion may be drawn from Moseley*s discovery that the L
lines progress in frequency from element to element just as do the K lines, the
frequency being in each case between one seventh and one eighth as great. It
is that, if there is a first or inmost electronic orbit, there must also be a second

^ This is the proper statement of the Moseley law, as he himself interpreted his results.
He knew and was careful to state that there is not an exact linear relation between the atomic
number and the square roots of the frequencies, but the lack of exactness of (2) both as to
straightness and to intercept may weU be attributed to secondary causes (see below).

* Verb. d. D. Phys. Ges., XVII., p. 152, 1913.

•Phys. Rbv., 6, 239, '15; also Phys. Rev., July, 1917.

Na'afi r^^ AMERICAN PHYSICAL SOCIETY. 1 99

one in all elements the radius of which is given by (i) to be about (8)^ or four
times as great as that of the first, provided orbital frequencies are in this case
too assumed proportional to observed frequencies.^

Guided then by the newly discovered facts of X-radiations, and the unques-
tioned laws of force between electric charges, we get our first information as
to the probable positions and conditions of some at least of the negative elec-
trons within the atom.

Again, having found the highest natural frequency which can come from
any element, viz., that from uranium, it is of extraordinary interest to inquire
where, according to Moseley's law (2), the highest frequency line of the K series
would fall for the lightest known element, hydrogen, whose nucleus should
consist of but a single positive electron. This is obtained as shown in (2) by
dividing the observed highest frequency of any element by the square of the
atomic number. The shortest wave-length given out by tungsten, atomic
number 74, is, according to de Broglie's measurements, .177 X lo'"' cm., and
according to Hull's measurements, .185 X io~^ cm. This gives for the shortest
wave-length which could be produced by hydrogen, according to de Broglie
.177 X 10"* X 74* * 97*9/iMt and according to Hull 101.3/1/1. This is as close
as could be expected, in view of the uncertainties in the measurements
and the further fact that Moseley*s steps are not quite exact, to the head
of the ultra-violet series of hydrogen lines recently discovered by Lyman
and located at 91.2 /i/i. There is every reason to believe, too, from the form
of Balmer's series, of which this is the convergence wave-length, that this
wave-length corresponds to the highest series frequency of which the hydrogen
atom is capable. li is fairly certain ^ then, thai this Lyman ultra-violet series
of hydrogen lines is nothing but the KX-ray series of hydrogen. Similarly, it is
equally certain that the LX-ray series of hydrogen is the ordinary Balmer
series in the visible region, the head of which is at 365 /t/t. In other words
hydrogen's ordinary radiations are its X-rays and nothing more. There is also
an M series for hydrogen discovered by Paschen in the ultra-red, which in itself
would make it probable that there are series for all the elements of longer
wave-length than the L series, and that the complicated optical series observed
with metallic arcs are parts of these longer wave-length series. As a matter
of fact an M series has been found for six of the elements of high atomic number.
Thus the Moseley experiments have gone a long way toward solving the mystery
of spectral lines. They reveal to us clearly and certainly the whole series of
elements from hydrogen to uranium, all producing spectra of remarkable simi-
larity, at least so far as the K and L radiations are concerned, but scattered
regularly through the whole frequency region, from the ultra-violet, where the
K lines for hydrogen are found, clear up to frequencies (92)' or 8464 times as
high. There can scarcely be a doubt that this whole field will soon be open to
our exploration. How brilliantly, then, have these recent studies justified
the predictions of the spectroscopists that the key to atomic structure lay in

^ This assumption is in this esse inconsistent with the simple form of Bohr's theory,
although that theory stiU gives the ratio of the radii of the first and second orbits z to 4.

200 THE AMERICAN PHYSICAL SOCIETY. [iSS£

the study of spectral lines. The prophets little dreamed, however, that the
study of spectral lines meant the study of X-rays. But now, through this
study, a sub-atomic world stands revealed to us in simpler form than could
have been imagined. For the atoms are now seen to be, in their inner portions
at least, remarkably similar structures, with central nuclei which are exact
multiples of the positive electron, surrounded in each case by electronic orbits
which have certainly, so far as the inner ones are concerned, practically the
same relations in all the elements, the radii of all these orbits being inversely
proportional to the central charge or atomic number.

So far nothing has been said about a quantum theory or a Bohr atom. The
results have followed from the known properties of assumed circular electronic
orbits combined with Moseley*s experimental law, and supplemented by the
single additional assumption that the observed frequencies of corresponding
lines from different atoms are proportional to the orbital frequencies. If they
suggest, however, that the experimental facts do not necessitate the quantum
theory for their more complete interpretation, the consideration of the energ>-
relations involved — these have been entirely ignored thus far — reveals at
once the futility of that hope, or of that fear, according to the nature of your
predilections with regard to theory of quanta. For the experimental facts
and the law of circular electronic orbits have limited the electrons to orbits of par-
ticular radii. But the energy principle does not permit them to be so limited
without a sudden or explosive loss of energy whenever the orbit is obliged to
change. Suppose, for example, that a cathode ray strikes the atom and
knocks out an electron from a particular orbit. When this or some other
electron returns from infinity to this orbit, it must in this act adjust its energy
to the only value which is consistent with this orbit and its characteristic
frequency. Hence in the act of readjustment it must radiate a definite quan-
tity of energy. Or again, suppose that the nucleus loses a beta ray through
the radioactive process. Every electronic orbit must then adjust itself to the
new value demanded by Moseley*s law. But this it cannot do if its energy is
conserved. The only way to permit it to do so is to let it radiate a definite
amount of energy in the act of adjustment. This suggests that each emission
of a beta ray by a radioactive substance must be accompanied by a whole series
of characteristic gamma rays corresponding to each changed orbit. The
emission of an alpha particle, on the other hand, would require an absorption
rather than an emission of energy, since its egress diminishes rather than
increases the nuclear charge. Perhaps this is why beta rays are always ac-
companied by gamma rays, while alpha rays are not so accompanied. This is,
however, a speculation which does not immediately concern us here. The
important conclusion, for the purposes of our present subject, is that Moseley's
facts and unquestionable mechanics, combined with our two assumptions of
circular orbits and radiation frequencies proportional in different atoms to
corresponding orbital frequencies, lead inevitably to the explosive emission of
energy in definite quantities accompanying orbital readjustments. And there
is nothing particularly disturbing or radical about this conclusion either, for

Na"^*] ^^^ AMERICAN PHYSICAL SOCIETY. 201

we have no basis for knowing anything about how an electron inside an atom
emits its radiation. The act of orbital readjustment would be expected to
send out ether waves. The only difficulty lies in the conception of the stable,
non-radiating orbits between which the change occurs, and whether or not we
can see how such orbits exist, the experimental evidence that they do so exist
is now very strong, and it is to further evidence for their existence, since that
is the main point to be established if this theory of atomic structure is to
prevail, that I now wish to direct your attention.

I have already mentioned the facts of magnetism and of the Zeeman effect
which support the orbital point of view. But the strongest evidence is found
in the extraordinary success of the Bohr atom, which was devised before any
of these Moseley relationships, which have forced us to the essential elements
of the Bohr theory,^ had been brought to light. Bohr, however, was guided
solely by the known character of the line spectra of hydrogen and helium,
together with the rapidly growing conviction, now dissented from, so far as I
know, by no pn-ominent theoretical physicist, that the act of emitting electro-
magnetic radiation by an electronic constituent of an atom must, under some
circumstances, though not necessarily under all, be an explosive process. To
show what is the character of this evidence, let us consider first what are the
essential elements in the Bohr theory and second what have been the accom-
plishments of that theory. Bohr's experimental starting point is the Balmer
series in hydrogen the frequencies in which are exactly given by

\ til* »»' /

(4)

fti having always, for the lines in the visible region, the value 2, and tit taking
in succession the values 3, 4, 5, etc. As previously noted, Paschen had already
brought to light a series in the infra red in which ni was 3 and nj took the suc-
cessive values 4, 5, 6, etc. Lyman's discovery, subsequent to the birth of the
Bohr atom, of an ultra-violet series of hydrogen lines in which ni is i and ns
takes the values 2, 3, 4, etc., is not to be regarded as a success of the Bohr atom,
but merely as a proof of the power of the series relationships to predict the
location of new spectral lines. To obtain an atomic model which will predict
these series relationships for the simplest possible case of one single electron
revolving around a positive nucleus, Bohr assumed

A. A series of non-radiating orbits governed by equation (i). This is the
assumption of circular orbits governed by the laws which are known to hold
inside as well as outside the atom.

B. Radiation taking place only when an electron jumps from one to another
of these orbits, the amount radiated and its frequency being determined by
kp ^ At -^ At, h being Planck's constant and A\ and At the energies in the
two stationary states.

This assumption gives no physical picture of the way in which the radiation

takes place. // merely specifies the energy relations which must be satisfied.

• N. Bohr., Phil. Mag., 26, i. and 476. and 857, 1913. Also 29, 332, 1915; 30, 394. ipiS-

202 THE AMERICAN PHYSICAL SOCIETY. ^S?

The principle of conservation of energy obviously requires that the energy
radiated be i4i — i4t. Also this radiation must be assigned some frequency f,
and Bohr placed it proportional to the energy because of the Planck evidence
that ether waves originating in an atom carry away from the atom an energy
which is proportional to v,

C. The various possible circular orbits for the case of a single electron rota-
ting around a single positive nucleus to be determined by T ^ ^rhn in which
T is a whole number, n is the orbital frequency, and T is the kinetic energy of
rotation. This condition was imposed by the experimentally determined
relationship of the frequencies represented by the Balmer series.

It will be seen that, if circular electronic orbits exist at all, no one of these
assumptions is in any way arbitrary. Each one of them is merely the state-
ment of the existing experimental situation. The results derived from them
must be correct if the original assumption of electronic orbits is sound. Now
it is not at all surprising that A, B and C predict the sequence of frequencies
found in the hydrogen series. They have been made on purpose to do it,
except for the numerical values of the constants involved. It was this sequence
which determined the form given to C. The evidence for the soundness of
the conception of non-radiating orbits is to be looked for then, first in the
success of the constants, and second in the physical significance, if any, which
att€Lches to assumption C. If the constants come out right within the limits
of experimental error, then the theory of non-radiating electronic orbits has
been given the most critical imaginable of tests, especially if these constants
are accurately determinable.

What are the facts? The constant of the Balmer series in hydrogen is
known with the great precision attained in all wave-length determinations and
has the value 3.290 X 10^*. From A, B and C it is given by the simplest
algebra as

^ " — ii— (5)

I have recently redetermined e^ with an estimated accuracy of one part in 1,000
and obtained again the value 4.774 X lo"^^ which I published in 1913.^ I
have also determined ^* h " photoelectrically' with an error, in the case of
sodium, of no more than .5 per cent., the value for sodium being 6.56 X 10"*'.
The value found by Webster* by the method discovered by Duane and Hunt'
is 6.53 X 10"*^. Taking the mean of these two results, viz.: 6,545 X lo"*'
as the most probable value, we get with the aid of Bucherer's value of e/m which
is probably correct to one tenth per cent. N = 3.294 X 10** which agrees within
a tenth per cent, with the observed value. This agreement constitutes most ex-
traordinary justification of the theory of non-radiating electronic orbits. It
demonstrates that the behavior of the negative electron in the hydrogen atom
is at least correctly described by the equation of a circular orbit. If this equa-

» R. A. Millikan, Proc. Nat'l Acad., April 1917.
«R. A. Millikan. Phys. Rbv., VII., 36a, 1916.

NS"af '] ^^^ AMERICAN PHYSICAL SOCIETY. 203

tion can be obtained from some other physical condition than that of an actual
orbit it is obviously incumbent upon those who so hold to show what that con-
dition is. Until this is done it is justifiable to suppose that the equation of an
orbit means an actual orbit.

Again, the radii of the stable orbits for hydrogen are given easily from Bohr's
assumptions as

In other words, since n is a whole number, the radii of these orbits bear the
ratios i, 4, 9, 16, 25. If normal hydrogen is assumed to be that in which the
electron is on the inmost orbit, 2a the diameter of the normal hydrogen atom,
comes out i.i X lo"*. The best determination for the diameter of the hydro-
gen molecule yields 2.2 X lo'"', in extraordinarily close agreement with the
prediction from Bohr's theory. Further, the fact that normal hydrogen does
not absorb at all the Balmer series lines which it emits is beautifully explained
by the foregoing theory, since according to it normal hydrogen has no electrons
in the orbits corresponding to the lines of the Balmer series. Again, the fact
that hydrogen emits its characteristic radiations only when it is ionized
favors the theory that the process of emission is a process of settling down to
a normal condition through a series of possible intermediate states, and is
therefore in line with the view that a change in orbit is necessary to the act
of radiation. Similarly, the fact that in the stars there are 33 lines in the
Balmer series, while in the laboratory we never get more than 12 is easily
explicable from the Bohr theory, but no other theory has ofTered even a sugges-
tion of an explanation. But while these qualitative successes of the Bohr atom
are significant it is the foregoing numerical agreements which constitute the
most compelling of evidence in favor of the single arbitrary assumption con-
tained in Bohr's theory, viz.: the assumption of non-radiating electronic orbits.

Another triumph of the theory is that the assumption C, devised to fit a
purely empirical situation, viz., the observed relations between the frequencies
of the Balmer series is found to have a very simple and illuminating physical
meaning, viz., the atomicity of angular momentum. Such relationships do not
in general drop out of- empirical formulae. When they do we usually see in
them real interpretations of the formulae — not merely coincidences.

Again the success of a theory is often tested as much by its adaptability to the
explanation of deviations from the behavior predicted by its most elementary
form as by the exactness of the fit between calculated and observed results.
The theory of electronic orbits has had remarkable successes of this sort.
Thus it predicts, as can be seen from 4, 5 and 3, the relationship which we as-
sumed, viz., that for corresponding lines (like values of ni and tit in 4) the or-
bital frequencies n are proportional to the observed frequencies v, and similarly
it predicts the Moseley law (2). But this latter relation, which is the only
one of the two which can be directly tested, was found inexact, and it should
be inexact when there is more than one electron is the atom, as is the case save

204 THE AMERICAN PHYSICAL SOCIETY. {

for hydrogen atoms and for such helium atoms as have lost one negative charge,
and that because of the way in which the electrons influence one another's
fields. It will probably be found to break down completely for very light
atoms like those of lithium. The more powerful the nucleus, however, and the
closer to it the inner orbit the smaller should this effect be. Now precisely
this result is observed. The Moseley law holds most accurately when tested
for hydrogen and the elements of highest atomic number and much less ac-
curately when tested for hydrogen and aluminum or magnesium. Similarly
the ratio between the frequencies of the a and ^ lines of the K series approaches
closer to the theoretical value (thai for hydrogen) the higher the atomic number
of the element.

Again, it is now well known that the a, /3, and 7 lines in the characteristic
X-ray spectra are not single lines as required by the simple theory. Accordingly
Sommerfeld* extended Bohr equations in the endeavor to account for this
structure on the basis of ellipticity in some of the orbits, and Paschen* by
measurements on the structure of the complex helium lines has obtained so
extraordinary checks upon this theory that ejm comes out from his measure-
ments to within a tenth per cent, of the accepted value.

A further prediction made by the theory and discovered as soon as looked
for was the relation between the lines of two succeeding series of this sort.

This should hold accurately from the energy relations between the orbits
whether there be one or many electrons in the atoms. I have been able to
find no case of its failure, though the data upon which it may be tested is now
considerable. I have also recently pointed out* that it is equivalent to the
well-known Ryd berg-Schuster law* which has been found to hold quite gen-
erally among optical series. Finally the ionizing potential of hydrogen is
given by Bohr's equations as 13.54 volts while experiment yields 11.5 volts.
This discrepancy is no way prejudice? the theory, but rather lends it support,
for the computed value is for the hydrogen atom while the observed value
relates to the hydrogen molecule which, in view of the repulsions between its
two negative electrons might be expected to be ionized more easily. Similarly
the computed value for helium which has lost one negative electron is 524 volts,
but the neutral helium atom is found experimentally to be ionized at the much
lower value 20.5 volts. That Bohr computed this latter value at 27 instead of
20.5 volts is not at all serious, since he had to make very particular assumptions
to obtain this result.

If then the test of truth in a physical theory is large success both in the pre-
diction of new relationships and in correctly and exactly accounting for 6ld
ones, the theory of non-radiating orbits is one of the well-established truths of

* Annalen der Physik, 51,1, 1916.

* Annalen der Physik, Oct., 1916.

■ Phys. Rev., May, 191 7, presented before Amer. Phys. Soc., Dec. i, 1916.
< Baly's Spectroscopy, p. 488.

Physical Reviev

Fig. 2. Showing absnrplion (in K r

R. A. MILLIKAN.

E.i'if--'"'

Fig. ,!. Showing ahsorplion (L region) bj- uranium and tliorium

R. A. MILLIKAN.

fSt^] ^^^ AMERICAN PHYSICAL SOCIETY. 205

modern physics. For the present at least it is truth, and no other theory of
atomic structure need be considered until it has shown itself able to approach
it in fertility. I know of no competitor which is as yet even in sight. I am
well aware that the facts of organic chemistry seem to demand that the valence
electrons be grouped in certain definite equilibrium positions about the peri-
phery of the atom, and that at first sight this demand appears difficult to re-
concile with the theory of electronic orbits. As yet, however, there is no
necessary clash. Hydrogen and helium present no difficulties, since the former
has but one valency, and the latter none. It is to these atoms alone that the
unmodified Bohr theory applies, for it treats only the case of a single negative
electron rotating about a positive nucleus. That the K radiations of the heavy
elements are so accurately predictable from those of hydrogen indicates indeed
that close to the nucleus of these elements there lie electrons to which the Bohr
theory fairly accurately applies, but the radiations give us no information about
the conditions or behaviors of the external electrons which have to do with the
phenomena of valency, and we have investigated but little the radiating proper-
tie? of the atoms which possess but few electrons. A further study of the be-
havior with respect to X-rays of the elements from lithium, atomic member 3,
to magnesium, atomic number 11, may be expected to throw new light on this
problem.

It has been objected too that th6 Bohr theory is not a radiation theory because
it gives us no picture of the mechanism of the production of the frequency p.
This is true, and therein lies its strength, just as the strength of the ist and 2nd
laws of thermodynamics lies in the fact that they are true irrespective of a mech-
anism. The Bohr theory is a theory of atomic structure; it is not a theory of
radiation, for it merely states what energy relations must exist when radiation,
whatever its mechanism, takes place. As a theory of atomic structure, however,
it is thus far a tremendous success. The radiation problem is still the most
illusive and the most fascinating problem of modern physics. I hope to discuss
it at a later time.

Amplification of the Photoelectric Current by the Audion.*

By Jakob Kunz.

THE photoelectric cell is used already as a photometer for many scientific
investigations, for instance in stellar photometry, in plant physiologyi
in researches on phosphorescence, on transmission, absorption, reflexion and
radiation of light in various forms. It would find further applications, even
for technical purposes, if the photoelectric current were larger. The writer
has some time ago constructed a photoelectric relay, where the primary photo-
electric current was increased considerably by a second electric field in the cell.
However this relay has not been studied and developed sufficiently and it was

' Abstract of a paper presented at the Washington meeting of the Physical Society, April
ao-3i. 1917.

2o6

THE AMERICAN PHYSICAL SOCIETY,

found that the audion, which has already so many important applications,
amplifies the photoelectric current to a higher degree than the relay. The
arrangement of the apparatus is given in the following figure, where P repre-
sents the photoelectric cell, Gi and Gt two galvanometers, Bu Bs, Bi batteries
of 120, 20 and 8 volts; A the audion. The photoelectric cell was illuminated

Fig. 1.

by an incandescent lamp of 2 candles at various distances from 30 to 170 cm.
from the cell ; the battery B^ supplies the heating current of the filament of the
audion. The galvanometer Gv of the primary circuit was 32 times more sen-
sitive than Gt of the secondary drcuit. The amplification of the photoelectric
current by the audion depends largely on the temperature of the incandescent
filament; the higher the temperature, the greater the secondary current, as
will be seen from the following readings.

34,5

102

34.0

130

180

250

500

When the deflection of the secondary galvanometer was 500, the temperature

of the filament was that of beginning white heat; the secondary deflection is

15 times larger than the primary, or the secondary current is 480 times larger

than the primary photoelectric current. And it was easy by raising the

temperature to amplify the photoelectric current 1000 times. With higher

temperatures the secondary currents become less steady. A small variation

of illumination produces a large variation in the secondary deflection. The

voltage applied to the photoelectric cell connected with the audion can be

raised above the point, where the glow in the cell sets in when independent of

the audion.

Laboratory of Physics.
University of Illinois,

Urbana, Illinois, April, 191 7.

SS'a^*] rflB AMERICAN PHYSICAL SOCIETY. 20/

High Vacuum Spectra from the Impact of Cathode Rays.*

By Louis Thompson.

EP. LEWIS, A. S. King and others have described experiments on the
• production of metallic spectra by cathodo-luminescence.
R. H. Goddard has devised a means' of increasing the light in one section
of the tube, due to the impact of cathode rays, thus making it i>ossible to work
with much higher vacua. The method consists essentially in
l^ the use of a magnetk field applied parallel to the axis of two small

I negatively charged plates, which also serve as cathodes. The

v^ y/ combined effect of the electric and magnetic fields is to cause the

ill I (ii electrons to assume a closed helical path in the very limited region
>^ TV between these plates. This results in a small but bright cylinder of
light (between A, Bin the figure) even at extremely low pressures,
Fig. 1. where without the field, the illumination would be so faint that

satisfactory exposures would be impossible.
In addition to the advantage of having sufficient light for comparatively
short exposures at high vacua, there are no temperature effects and the pres-
sures are definitely known, especially when working with substances which
are normally gaseous. The use of capillary tubes for high vacuum work, while
possible, gives spectra which depend largely upon temperature and current
density.

At high vacua and with temperature effects absent, simple spectra are likely
to result. A recent paper by Mallik and Das' emphasizes this point.

A number of substances are being examined. The first exposures were
made with mercury vapor and tellurium, the mercury at pressures down to
that corresponding to a three-inch parallel spark.

Compared with the ordinary vacuum tube discharge the spectrum is simple
and the relative intensities are noticeably different. The important series
lines are prominent in the visible region: In the ultra-violet only 3131, 3125
and 2536 are strong. 2536 is the brightest line in the spectrum and seems to
become relatively stronger as the vacuum increases.

Experiments are being made with hydrogen in an attempt to determine
whether at high vacua and with the increased illumination, the extended
Balmer series, which has only appeared in certain stellar spectra, can be found
in a discharge tube.

The magnetic field used in the work is not strong enough to produce a
detectable eflFect on the spectrum.

Clakk Untvbrsity.
WoRCBSTER, Mass.

> Abstract of a paper presented at the Washii^ton meeting of the Physical Society. April
ao-2i. 1917.

* Described in U. S. Patent 1.137,964.

* Phil. Mag., March. 1917.

208 TBB AMERICAN PHYSICAL SOCIETY. |^S2

A Proposed Method for the Photoubtry of Lights of Diffbkbnt Colors.

—111."

By IRWIM G, Pbikst.

TWO papers under the above title* have already been communicated to the
Physical Society by the author. The purpoaes of the present paper are:

1. To present some data on spectral distribution in order to further elucidate
and substantiate the method.

2. To publish, for reference, a table of numerical values of the method factor
mentioned in the first paper and explicitly defined in the second paper as

VE .in" (* - «)<ft

VEd\

3. To note some advantages and possible applications of the method.

I. The Relative Spectral Distribution of Lights Color-matched by the Proposed
Method. — The essential feature of the method is that light from a comparison
source of known spectral distribution is modified by passage through a quartz
plate between nicol prisms so as to match the color of the source being tested.
The quartz plate and the nicols constitute, in elTect, a " color screen " of readily
adjustable spectral transmission, the relative transmission for any wave-length
being given by*

8in» (* -a).
Typical curves showing values of this expression as a continuous function of
X for various values of 0 (" spectral transmission curves ") are shown in Fig. i.

Fig. 1.

Relative transmUaion of a ayscem compoaed of a i tutu, plate of quaiti between niool
prisms, a — rotation of plane of potarization by the qiuuti ptate (a function of wave-
length); ^ — rotation of analyzing nicol from position for extinction (quarti removed).
(Both a and 4 are measured cloclcwiBe from obKrver's poaition).

1 Abstract of a paper presented <by title) at the Washington meeting of the Physical So-
ciety, April. igiT.

'Am. Phys. Soc., Wash.. April, 1515T Phys. Rbv. (2), 6. 64, Am. Phy», Soc., N. Y..
Dec, 1916; Phvs. Rbv. (i). g, 341 (for erratum see p. sBo).

■ For definition of symbols see Phys. Rbv., 9, 343.

Naa. J

THB AMERICAN PHYSICAL SOCIETY,

209

In the previous papers It was merely stated that a sensibly perfect color-
match had been obtained. Fig. 2 in the present paper compares the spectral

• iaa

W*1

0 c

0 4

Fv*"

■WMI

><»<l

iCfilM.

SmmM

»»w

• «

• .4a

>«#fa<«

(CM

>£^

.:.akA.«i

Ph«

».'Aur-<i;

■

•

^__

« ^*

Fig. 2.

Two spectral radiation distributions giving an empirical color match in the relative candle-
power determinations reported in paper No. II. of this series.

distribution in light from the carbon lamp at 4 w.p.m.h.c.^ with the spectral
distribution of the light indirectly made to color-match it by the proposed
method in the determination of relative candlepowers reported in the second
paper cited above.

Fig. 3 shows the theoretical possibility of color-matching, by this method,

"^^

■

X>4

• • •

tj-.*

OS"'

Mrfi

•w»i

*Jhi

iM^

< 1

c ^ .

•

I
•

1 "

•

,•

"»-

K -

. 1

•
\

• .

«*•

•1

a»

Fig. 3.

Illustration of the matching of a given spectral radiation distribution by a distribution
computed by the rotatory dispersion method.

the light from a gas-fUled tungsten lamp. This does not represent an em-
pirically determined color-match, but merely compares the spectral distribution
of the gas filled lamp,* with that of acetylene as modified by the quartz-nicol
system, the thickness of quartz and the angle 0 having been theoretically chosen
to give the desired spectral transmission. The exact efficiency of the gas-filled
lamp in this case is not known, but was probably about 15 lumens per watt.

^ Determined radiometrically by W. W. Coblentz and communicated to the author-
* Radiometrically determined by W. W. Coblentz.

2 1 0 THE AUBmCAN PHYSICAL SOCIETY. lS»

Fig. 4 compares the radio metrically determined spectral distribution' of
a gas-filled lamp at 15.6 lumens per watt' with the spectral distribution of the
light empirically found to color-match it by this method, the match being
made by the author using the Arons Chromoscope.*

The dotted and dashed curves in Fig. 5 show the spectral distributions of

Fig. 4.

lights found, by this method, to color-match a gas-filled lamp at 33.3 lumens
per watt and at 32.o lumens per watt.*

The solid curve from Luckiesh, " Color," page 21, is given for comparison.

When it is considered that the efficiency of a gas-filled lamp can not be taken
as a reliable specification of its color, and that the photometry of these lamps
is a matter of considerable uncertainty, this discrepancy appears hardly greater

Fig. S.

than might be expected. It will be noted that the agreement of curves in Fig. 4,
where the data are strictly comparable (pertaining to the same lamp), is con-
siderably better.

It will be understood, of course, that the author has no idea of here testing

' By Coblents and Emerson, Bureau ot Standard!.

■ By Crittenden and Taylor. Bureau of Standards.

'Ann. der Pbys. 39. 545. 1913.

* By Crittenden and Taylor, Bureau of Standards.

^^] THE AUERICAtr PHYSICAL SOCIETY. 2 1 I

the accuracy of spectral distributions determined radio metrically or spectro-
photometrically. The purpose of presenting this data is merely to show how
closely this method of empiric color-matching does, in fact, give a spectral match.
Those familiar with the subject will readily see that this method of color*
matching gives better spectral matches than are obtained by colored glasses,
which have been used for the same purpose.

A given source may be color-matched using quartz plates of different thiclc-
aesses, each plate, of course, requiring a different value of 0 to give the color-
match. For example, with acetylene as a comparison source, a gas-filled tung-
sten lamp at 13. 3 lumens per watt may be color-matched with the following
thicknesses of quartz and corresponding values of ^:

TblckatM, mm. 4, dafraai.

0.50 162.4

.75 1S7.9

1.00 IS4.6

The spectral distribution curves from these three sets of data are much the
same; but have, indeed, slightly different curvatures, so that one of them
may match the true spectral distribution of the lamp better than the others.
Further consideration is being given to the choice of the quartz thickness best
suited to give a very close spectral match where that is desired, but i.ooo mm,
appears very satisfactory for the photometry of the incandescent lamps now
in use (carbon, vacuum tungsten and gas-filled tungsten).

2. Table of Ike Method Factor. — In the previous papers, it was pointed out
that computations in terms of this method, while apparently quite involved,
could be made very simple by the use of permanent reference tables of the
" method factor," R. The labor of preparing such a table, based on the most
recent and reliable data for " visibility " and, " spectral distribution of radiant
power " has now been completed,* and the table is presented herewith in the
hope it will be useful to any who care to investigate or use the method.

For illustrative purposes, the form of the function R is shown in Fig. 6.

Fig. 6.

All steps in the preparation of this table have been carefully checked and the
author trusts that it will be found reliable, fie hopes however that some one
may find interest enough in the subject to check this table entirely independently
assuming nothing except the same fundamental data for £, V and a, and the
arbitrary values for 0.

I Graphic tntegiatloD and computation by H. J. McNicholaa. J. T. Filiate and H. E. Cole.

2 1 2 THE AMERICAN PHYSICAL SOCIETY. [SS

As the subject develops, it may be later found desirable to know R for some
particular values of 0 more accurately than it is given in the present table.
The siuthor hopes to undertake such determinations later. Before this method
can be generally adopted, it will of course be necessary for those interested and
in authority to agree upon standard values for V and E as used in the formula
for R. Instead of taking E for acetylene, it may be best to refer directly to a
black body at some specified temperature.

3. Notes on Advantages and Applications of the Method. — (a) It is evident
from the data presented above that this method is superior to the color glass
method from the point of view of obtaining a close spectral match as well as a
** color- match."

(b) Selective transmission obtained and controlled in this way is universally
reproducible and subject to fine adjustment, while that obtained by glasses,
is, in general, neither.

(c) Spectral transmission determined by this method is more reliable than
that of a glass determined by means of the spectrophotometer; and, by use of
permanent tables, may be more readily obtained.

(d) The data obtained in determining candlepower may be readily translated
into a significant specification of the color of the light being tested.

{e) The method is applicable to the production of " artificial daylight "
from tungsten lamps, giving a much better spectral match than has been
obtained by the use of blue glass. The author is preparing a separate paper
on this subject.

(/) At the request of Dr. N. E. Dorsey, application of this method to de-
termine the brightness of very faint greenish fluorescent screens is being at-
tempted. A satisfactory color-match has been found.

ig) It is obvious that the photometer described in the preceding paper in

this series, may be used as a pyrometer, and might be calibrated to read

temperatures directly, but the author ventures no opinion as to its value for

this purpose.

National Burbau of Standards.
April 17, 1917.

No. a. J

THE AMERICAN PHYSICAL SOCIETY.

213

THE METHOD FACTOR R AS A FUNCTION OF 0

Computed by H. J. McNicholas. J. T. Filgate and H. E. Cole, Dec. 1916-Jan.. 1917.

r^ VE 8in« (0 - a)dK

VEdK

assuming the following functions of X:

V -■ Relative Visibility, Coblentz and Emersont Phys. Rev. 9, 88.

E ■■ Relative Radiant Power Acetylene Flame. Coblentz & Emerson. B. S. Bull 13, 363.

a ■> Specific Rotation of Quartz (Degrees/mm). Landolt. Op. Dreh. and Ed., p. ia8;
and Landolt BOmstein Tables. 1913 Ed., p. io6a.

(This table was constructed by graphic interpolation from values determined by graphic
integration at intervals of 5°. The values tabulated are means of two independent inter-
polations. They are considered reliable to about 0.00 1.

♦.

If.

♦.

If.

1»

0.1425

46»

0.1545

910

0.856

136»

0.844

132

1675

867

137

831

121

1805

879

138

818

1105

194

889

139

805

1005

209

899

140

791

091

222

9095

141

777

081

2365

9195

142

7615

072

251

9285

143

747

064

266

938

144

7315

056

2815

100

946

145

716

048

297

101

954

146

700

041

3135

102

9605

147

6845

034

330

103

9675

148

668

028

347

104

973

149

652

023

364

105

978

150

636

018

3805

106 .

983

151

620

014

397

107

9865

152

6035

Oil

4135

108

990

153

586

008

4305

109

993

154

570

006

447

110

995

155

552

004

4645

111

9955

156

5355

004

482

112

9955

157

518

0035

4995

113

995

158

5005

004

5165

114

994

159

4825

005

534

115

9925

160

4645

006

551

116

991

161

448

008

5675

117

9885

162

430

0115

585

118

9865

163

413

0145

6015

119

984

164

3955

5185

6185

120

980

165

378

0235

6355

121

9765

166

361

029

652

122

9715

167

345

0355

6685

123

9665

168

329

042

685

124

960

169

3125

049

7015

125

953

170

2965

056

717

126

9465

171

2805

064

733

127

9385

172

266

072

7485

128

9305

173

2505

081

763

129

922

174

236

090

7775

130

9125

175

222

099

7915

131

902

176

2075

109

8055

132

891

177

1935

1195

819

133

8805

178

1805

1305

8325

134

8685

179

167

142

8445

135

856

180

1545

214 ^^^ BOOKS. [i

LSBEn&

NEW BOOKS.

A System of Physical Chemistry. By William C. McC. Lewis. New York:

Longmans Green and Company, 191 6. In two volumes. Vol. L, pp. xiv

+523; Vol IL, pp. vii+552.

As this book belongs to the well-known series of text-books on physical
chemistry which includes Findlay's ** Phase Rule," Mellor's ** Chemical Statics
and Dynamics," and Young's " Stoichiometry," one naturally expects a high
standard of excellence. Although opinions niay differ as to whether the
present work deserves as high a rating as the others just mentioned, there can
be no doubt that the author has performed a difficult task with remarkable
success. In respect to completeness and extent of ground covered, especially
in the more recent developments of the subject, this book is not equalled, so
far as known to the reviewer, by any other text-book of physical chemistry
originally written in English.

Some previous knowledge of the subject is assumed and a few of the topics
emphasized in the average text are omitted, or at least not separately discussed.
We look in vain for sections on thermochemistry, the periodic system, deter-
mination of atomic weights, relations between physical properties and chemical
constitution, and several other familiar titles. On the other hand the author
has aimed to include " some account of recent investigation " and cites in the
preface, as examples, the sections dealing with the structure of the atom, the
theory of concentrated solutions, capillary chemistry, Nernst's theorem of
heat, the thermodynamics of photochemical reactions, and the application of
the Planck- Einstein ** Energie Quanta " to the specific heat of solids. How
thoroughly these subjects are treated can be inferred from the fact that about
125 pages of text are devoted to the last three alone. In general, the author
has been very conscientious in including and giving ample space to the newer
developments of the subject, but there are some exceptions. For example, in
dealing with the structure of the atom the theories of J. J. Thomson and of
Nicholson are each treated at length, while the much more important theory of
Rutherford is barely mentioned in a foot-note. It is also surprising, and very
much to be regretted, that in so large a book a few pages have not been devoted
to the recent classification of the radio-elements according to chemical proper-
ties and atomic numbers, and to the conception of isotopes, subjects of great
importance and exceptional interest.

The arrangement of the subject matter is novel. Volume I. is entitled
" Kinetic Theory " and Volume II., " Thermodynamics and Statistical Me-
chanics." Underlying this arrangement, which we are told is merely arbitrary
and intended to make the book more readable, is the author's system of classi-
fication which is presented in tabular form at the beginning of the book. The

X^jf-] NEW BOOKS. 215

actual arrangement of the material seems to be a sort of compromise between
these two divergent plans of classification and as a result many topics treated
in the first volume have to be taken up again in the second. This system is
confusing to the uninitiated, and its advantages, to say the least, are not
apparent.

In general, the author presents his subject matter with commendable clear-
ness, but perhaps overemphasizes his lack of personal bias, and the statements
of more or less conflicting views are not always accompanied by any actual
sifting of the evidence. This, and the peculiarities of arrangement, seem to the
reviewer the most evident faults in an otherwise excellent and useful book.

R. G. V. N.

Electrical Engineering, Advanced Course, By Ernst Julius Berg. New York:
McGraw-Hill Book Company, Inc., 1916. Pp. viii+332.

The author has made a constructive contribution in the field between the
more usual treatment of alternating currents, as given by Steinmetz, Bedell
and Crehore and others, and advanced electrical theory as given by Maxwell,
Heaviside, Webster and others. Although he has not given us a complete
bridge between the two, he has supplied some helpful stepping stones and has
performed a useful service.

F. B.

Color and its Applications, By M. Luckiesh. New York: D. Van Nostrand
Co., 1915. Pp. xii+357. Price, J3.00.

In the words of the author: ". . . the desire has been frequently expressed
for a book that treated the science of color as far as possible from the viewpoint
of those interested in the many applications of color.'' The object of the
present volume b to meet this demand.

The opening chapters are devoted to a discussion of fundamental concepts
concerning the nature of h'ght and color. This is followed by a treatment of
color analysis and synthesis and allied subjects The final chapters deal with
the practical applications of the preceding theoretical considerations. Such
subjects as Color in Lighting, Color Effects for the Stage and Displays, Color
Phenomena in Painting, Color Matching, Art of Mobile Colors and Colored
Media are discussed.

Considering the book as a whole, it serves well the purpose for which it was
written. Since its appeal is to the general public, the author has avoided
wisely the use of mathematics. While a number of misstatements occur in
the earlier chapters, the treatment is, as a rule, clear. Numerous cuts and
half-tones illustrate the text admirably. The later chapters, which deal with
practical applications and which embody much of the author's own work, form
by far the most important part of the book.

By the choice of large type and unglazed paper, the author has given a
unique application of some of the principles laid down in the text.

A. H. P.

2l6 NEW BOOKS. [to?S

Concise Technical Physics, By J. Loring Arnold. New York: McGraw-
Hill Book Company, Inc., 1916. Pp. viii+275. Price, J2.00.

To quote from its preface: ** This book is an embodiment of actual class-
room work in the first college course in theoretical physics. It is intended
primarily for use in schools of engineering, but may be used in other collegiate
institutions equally well." The title raises false hopes, for the interesting
but very difficult problems of technical physics are not the subject treated.
The book should be called *' Physics for Engineering Students." Numerous
instances of the author's lack of a thorough command of his subject prevent
the reviewer, much to his regret, from recommending this book.

E. B.

Principles of Alternating Current Machinery, By Ralph R. Lawrence.
New York: McGraw-Hill Book Company, 1916. Pp. xvii+614.

This book is not a description of alternating current machinery, nor is it a
treatise on the theory of alternating current; its subject matter is exactly as
indicated in the title — the principles of alternating current machinery. The
author discusses synchronous generators, static transformers (the adjective,
but not the treatment, is quite out of date), synchronous motors, parallel
operation of alternators, synchronous converters, polyphase induction motors,
single phase induction motors, series and repulsion motors. The discussion
is clear and is given with great thoroughness — almost exhaustive thoroughness
so far as principles go. The book will prove useful both to student and

engineer.

F. B.

Preliminary Mathematics. By F. E. Austin. Hanover, N. H., 191 7. Pp.
i-f 169. Price J1.20. {Received.)

Analytic Geometry. By W. A. Wilson and J. I. Tracey. New York: D. C.
Heath and Co., 191 5. Pp. x+212. {Received.)

A Chemical Sign of Life. By Shiro Tashiro. Chicago: The University of
Chicago Press, 1917. Pp. ix+142. Price, Ji.oo. {Received.)

The Biology of Twins {Mammals). By Horatio Hackett Newman. Chi-
cago: The University of Chicago Press, 191 7. Pp. ixH-i89. Price, $1.25.
{Received.)

Finite Collineation Groups. By H. F. Blichfeldt. Chicago: The University
of Chicago Press, 1917. Pp. xi + 193. Price, J1.50 net. {Received.)

Lemons sur Les Fonctions Elliptiques en vue de Leurs Applications. By R. de
MoNTESSUE DE Ballore. Paris: Gauthier-Villars et Cie, 1917. Pp. x +
267. Price, 12 fr. {Received.)

A Laboratory Course of Practical Electricity for Vocational Schools and Shop
Classes. By Maurice J. Archbold. New York: The Macmillan Com-
pany, 1 91 6. {Received.)

Second Series. September, igi7 Vol. X., No. 3.

THE

PHYSICAL REVIEW.

on;a general expansion theorem for the transient
oscillations of a connected system.

By John R. Carson.

IN the usual solution of the problem of the transient oscillations of a
connected mechanical or electrical system in response to a suddenly
impressed set of forces, the determination of the characteristic modes of
oscillation (periodicities and dampings) is comparatively easy, since it
involves only the determination of the roots of a polynomial. As regards
the amplitudes of the transient oscillations the case is different. The
usual procedure is to designate the amplitude of each mode of oscillation
of each coSrdinate of the system by an undetermined constant, sub-
stitute in the equations which describe the system, and then determine
the unknown constants in accordance with the given initial configuration
of the sysem. This method of determination, while perfectly straight-
forward, is extremely laborious, and the difficulty increases rapidly with
the number of degrees of freedom of the system. When the initial con-
figuration is arbitrary no other method than that outlined above is known
to the writer; when, however, a set of forces is impressed on a system at
rest or in equilibrium configuration the amplitudes of the transient oscil-
lations admit of much simpler determination by the expansion theorem
developed in this paper.

So far as the writer is aware no one, with the exception of Heaviside,
has attacked the problem of a general formulation of the transient
oscillation as regards their amplitudes as well as periodicities. Heaviside
in his Expansion Theorem^ gave a very valuable formulation of the
transient oscillation of an electrical network when the oscillations are
excited by the sudden application to the system of an electromotive force
which is not a function of time; that is a steady uniform electromotive
force.

'See Heaviside, Electromagnetic Theory, Vol. II., p. 137.

217

2 1 8 JOHN R. CARSON. [i

In the present paper the general solution of the problem considered by
Heaviside will be developed and an Expansion Theorem derived which
formulates the resultant (forced and transient oscillations) of a connected
dynamical system in response to an arbitrary driving force applied to
any co5rdinate of the system. The only limitation imposed on the form
of the driving force is that it shall be capable of expansion in a Fourier's
integral or else expressible as a complex exponential function of time,
a limitation which constitutes no restriction of practical importance.
The Expansion Theorem to be derived is thus of broader application
than Heaviside's Theorem which constitutes a particular case of the
general theorem;* furthermore its derivation may be of interest since
Heaviside states his theorem without proof.*

The dynamical system to be considered may be either a mechanical or
electrical system characterized by a symmetrical system of linear dif-
ferential equations of the following form:*

aiixi + aiiXt + ai^z + • • • + amXn = Fu

atiXi + atiXt + at^z + • • • + atnPCn = Fj, . .

In equations (i) Xu xt, *- * Xn are the displacements from equilibrium
or zero configuration of the n codrdinates Xu Xtt • • • -X"* specifying the
system while Fu Ftt • • • Fn are impressed forces. The general coefficient
ajk is of the form

<? d

where g/*, r,* and 5/jk are constants. The q coefficients will be termed
the inertia factor, the r coefficient the resistance factor and the s coef-
ficient the stiffness factor. From a mathematical standpoint no limit
is placed on the order of a^k in (d/di) ; it is only necessary that it be capable
of expression as a polynomial of the nth order in (d/dt). However in
cases of practical importance the polynomial a^k is of the second order
in d/dt as indicated by equation (2).
To simplify the following work it will be assumed that only one driving

* In El. Th., p. 131-2, Vol. II., the case of simply periodic forces is treated by the operationa
method.

* Since the above was written Mr. H. W. Nichols has called my attention to the fact that
Heaviside derives his Expansion Theorem in his Electrical Papers. Vol. II., p. 373. Mr. H.
W. Malcolm in a series of papiers entitled "The Theory of the Submarine Telegraph" appear-
ing in the Electrician during 19 12 attempts to prove the Heaviside Theorem. The method
of derivation is, however, quite defective.

* See Webster's Dynamics, 2d edition, pp. 173, 174.

VOL.X

No. 3-

EXPANSION THEOREM FOR TRANSIENT OSCILLATIONS.

219

force Fi is impressed and consequently Ff , Ft, • • • F, are put equal to
zero. It may be readily shown that this simplification involves no loss
of generality whatsoever, since the complete solution may be built up at
once from the formulae to be derived.
The driving force is assumed of the form^

Fi = i{£i€^' + £i€^M (3)

= R{Ei€^'] (4)

where E and p are constants. In formula (3) the bar denotes the con-
jugate imaginary of the unbarred symbol, while in formula (4) R indicates
that the real part of the expression alone is to be retained. For con*
venience the symbol R will be omitted and it will be understood that the
real part of the final expression is the solution.

The forced oscillations of the system are gotten by the well-known
method* of replacing d/dt by p in (i). If yu Vit • • • Vn denote the
forced components of Xu ^2, • • • x», then :

In formula (5), D{p) is the value of the determinant

(5)

ail an ai8

aai dti ajs

asi ass

... ...

(l2f

dZt

(6)

fliil flnl flni • • • flnn

when the operator d/di is replaced by p. Mikip) is the minor of the first
row and Jfth column of D{p), The solution for the forced oscillations
is of course well known.

The complementary solution of equation (i) gives the transient oscil-
lations. If Zk denotes the transient component of xa, then Zu is ex-
pressible as:

Zk = E^*^"'^ • ^^' (7)

where pm is a root of the equation D{p) =0 and Ak^"^^ is an integration
constant to be determined by the connections of the system and the
initial configuration at time / = o. The summation is extended over the
roots of Dip) = o.

^ When the driving force is the arbitrary time function /(/) it can of course be expressed as
a Fourier Integral or Series, each of whose components is of the form given by (4) when p is
a pure imaginary. The explicit treatment of this case is reserved for a future paper.

* Loc, cU,

2 20 JOHN R, CARSON, [sewm.

The solution is then:

MkH

x» = y. + «. = E, ^^ «"+ E/t*"' • *'-'• (8)

In general the conditions to be satisfied by the solution are as follows:

1. The initial displacement of every coordinate shall be zero; that is
xjfc = o at / = o for all values of k.

2. The initial velocity of every co5rdinate shall be zero; that is

dxk

at / = o for all values of k.

3. The ratio of Zj to Zu for the mth mode of oscillation shall be equal
Mij(Pm)/Mik(Pm) . This last condition may be readily seen to be necessary
by substitution of (7) in (i), and is perfectly general. The first two
conditions follow from the fact that the initial configuration is one of
equilibrium. Certain particular cases when these conditions do not
hold are examined below.

We shall now proceed to a determination of the integration constants
of (7). The initial value (/ = o) of y* is by (5):

Mikip) . ^

(y*)i-o = ^^~D{i) • ^^^

Now from equation (2) and formula (6) D(p) is in general a polynomial
of the 2nth order in p while Mik{p) is a polynomial of the (2n — 2)
order in p. The right-hand side of (9) may be expanded by means of
the following theorem:*

If Qi^x) and P(,) are polynomials in x and if Po is of higher order than
Qix)> then:

P{x) i^, {X - x^)P\x^) ^^""^

where X^ is a root of P(,) = o, and

provided P{») does not contain repeated roots. The special case of
repeated roots will be briefly discussed later.

In general Mikip) and Dip) satisfy the conditions of expansion, whence

(12)

(yk) i«o = -til ~K/T^~ = -til 2^ 7- T~\w

Dip) ^i^x{p-p^)D'ip^)'
> See Williamson, Integral Calculus, pp. 43. 43.

NoT^'] EXPANSION THEOREM FOR TRANSIENT OSCILLATIONS. 221

where the summation is extended over the roots of D(p), and
Clearly then, both conditions (i) and (3) are satisfied if we set

since then (y» + «*) «_o = o and*

i4»<-> Mikipm)

(14)

Hence the complete solution is

^'-^^-dW' S^(p-p.)D'(py\^ (^5)

provided this solution satisfies condition (2). That this is in general
the case may be readily shown. Differentiating (15) we have

and

Now pMin,) is in general a polynomial in p of lower order by one than
D(p) whence

pMuip) _ "f? pn.Mn(p^)
D{p) ~ ^,ip - P^)D'{p^y ^'^^

SO that condition (2) is satisfied.

It is now easy to extend formula (16) to the more general case when
all the forces Fi • • • Fn are finite. For let

F, = £i€''; -^ Fn = £•€^^

Then the complete solution is

'* =^ S^'"^(^* - 5£^'(/>-/>.)i>'(^o* •

Of course the different forces may be characterized by different expo-
nential factors.

The conditions necessary that the partial fraction expansions given
by (12) and (16) shall hold are satisfied in general; that is in the usual

> Equation (14) is equivalent to condition (3), and formulates the necessary relation
among the constants of integration.

222 JOHN R. CARSON. ^SS,

case when the inertia and stiffness factors q and 5 are all finite. No
attempt will be here made to rigorously discuss the cases when the general
expansion fails or when it must be specially interpreted. Two physically
interesting cases will however be considered.

1. Assume that the inertia factors (q) are all zero. It will be clear
then from physical considerations that condition (2) will not necessarily
hold since finite velocities may be instantaneously established owing to
the absence of inertia. The initial configuration of the codrdinates must,
however, be zero from physical considerations. We should, therefore,
expect, from purely physical considerations, that the expansion given
by (12) is still valid while the expansion given by (16) no longer holds.
This is precisely the case since now Mik(p) is of order (n — i) and D(p)
of order n in p. Hence while the expansion of Mik(p)/D(p) is valid the
expansion of pMik(p)/D(p) is no longer valid since pMik(p) is of the same
instead of lower order than D(p). Thus while the expansion formula
(15) for the coordinates and consequently the expansion formula follow-
ing for the velocity are correct, the initial velocities are no longer neces-
sarily zero.

2. Assume that the stiffness factors (s) are all zero. Then physical
considerations show that an equilibrium configuration of the codrdinates
is indeterminate but that the initial velocities are necessarily zero. We
should therefore expect a priori that the expansion (12) is not necessarily
true but that expansion (17) is still valid. This is precisely the case as
results from the following consideration. If the stiffness factors (5)
are all zero, zero is a repeated root of D(p) of the nth order and a repeated
root of Mik(p) of the (n — i)st order. Then

pM.kiP) ^ P-Qip) ^ Q(^
Dip) p-Pip) Pip) •

where Q{p) and Pip) contain no zero roots. Then

pM.kip) ^ 'g Q{p^)

Dip) i^^iP" p^)P'ip^) '

when the summation is taken for all the roots of Dip) exclusive of zero.
It may then be readily shown that

QiP^) ^ pmMikip^)

(P - PndP'iP^) iP - P^)D'iP^) '

whence it follows that the expansion for the velocities is valid. The ex-
pansion for the co5rdinates is meaningless.

The two foregoing particular cases serve to illustrate the fact that
while the expansion is generally valid it will not hold for dynamic systems

Ko^^'] EXPANSION THEOREM FOR TRANSIENT OSCILLATIONS. 223

in which the initial conditions are not necessarily satisfied. If, therefore,
the expansions (12) and (16) are not valid we may be sure that the initial
conditions are not complied with by the system under consideration.
Further elaboration of this point is not believed necessary and particular
cases can be readily worked out from the general theory.

As staged above, the partial fraction expansion of equation (10) does
not hold when the denominator P(x) contains repeated roots. Cases,
however, in which the characteristics determinant of the system contains
repeated roots can be readily handled by letting the roots approach
equality as a limit. A brief example will suffice to indicate the appro-
priate treatment. Assume that the characteristic determinant is

D(p) = />« + 2a/) + a*

and let

Dip)

y = T^TIx «'*•

The roots of D(p) are then equal so that pi = pi = — a.
To handle this problem consider the general case where

D(p) = (/) - pi)(p - />,).
Then

Dip) ip - pi)ipi -pi) ip- p%)ipt - pi) '

Now let pi = — a + ^, pi = — a and let e approach zero as a limit.
The final expression for x is, in the limit

For the sake of generality the foregoing formulae have been derived in
terms of a general dynamic system; since, however the most important
application of the expansion theorem is concerned with oscillations of
electrical networks, the formulae will therefore be translated into the
terms of such a system. In formula (16) replace i» by /*, where /* is
the current in the Jfth branch or mesh of the network; let g, r and ijs
be inductance, resistance and capacity and let Zihip) be the ratio of the
E.M.F. of frequency p impressed on branch or mesh i to the forced current
flowing in branch or mesh K. Clearly Ziuip) may be termed the im-
pedance of the Kt\i with respect to the first branch and is given by

^''^^ - pMu(P) •

224 JOHN R, CARSON. ]SSmi

Also

since Dip,^ = o.
Formula (i6) may then be replaced by

where pw is a root of Ziuip) since the roots of Zi*(/>) are likewise the
roots of -D(/>). Formula (i8) is the generalized form of Heaviside's
Theorem, into which it degenerates when p is put equal to zero.

The expansion formula gives explicitly the resultant oscillations when
a driving force is suddenly impressed on the system. It may be also
used to formulate the subsidence to equilibrium of a system having any
initial configuration, provided such configuration is producible without
changing the connections or constraints of the system. This limitation
is equivalent to the statement that the initial configuration may be
formulated by sums of expressions of the form :

when T is to be regarded as a constant. The free oscillations back to
equilibrium are then given by

The expansion theorem formulated by (i8) is derived in terms of system
which is specified by a finite number of coordinates. That it holds for
a system characterized by an infinite number of co5rdinates is a fair
inference, since it seems permissible to let the number of co5rdinates
approach infinity as a limit, though doubtless a rigorous proof of this is
necessary. However the Expansion Theorem does hold for a number
of problems involving an infinite number of co5rdinates which have been
examined by the writer; in particular the Expansion Theorem may be
applied to the oscillations of a transmission line having distributed con-
stants as well as to an artificial line having a finite number of lumped or
localized elements.

To illustrate the application of the Expansion Theorem to the oscilla-
tions of a transmission line, assume an electromotive force expressible as

Na*^*] EXPANSION THEOREM FOR TRANSIENT OSCILLATIONS, 225

i? {£€***} to be impressed at time / = o on a transmission line of induc-
tance L, capacity C, resistance R and leakage G per unit length. Let the
length of the transmission line be / and let the e.m.f. be impressed through
an impedance Zi at 5 = o which the line is closed by an impedance Zt
at 5 = /. The ** forced " component current at point s on the line, cor-
responding to the impressed e.m.f. is then expressible as

where

, K{Z, + ZQ + (JP + ZiZQ tanh (yf)

'^'^' " cosh {ys)[KiZi + Z,) + {K} - ZiZ,) tanh (yl)] ' ^^°^

— sinh ys[Zt + K tanh {y)t[
In the foregoing formula:

y^<{R + Lp]\G + Cp], (21)

--^[

^^^ } (22)

Z\ and Z« are, of course, preassigned explicit functions of p.
In accordance then with equation (i8) the expression for the current
at any point s along the line, valid for positive values of t, is

^' ~ ^ I H>.{p) h ip - />«)♦>.'(/>«) J • ^^^^

where <p»{p) is given by (20) ; pm is the fwth root and <p»'(p) is the derivative
of fp»(p) with respect to p, and the summation is extended over all the
roots. There are of course an infinite number of roots of the transcen-
dental function <p$(p) so that in general the solution is practically un-
manageable. It is however, a formal compact solution of the problem.
Moreover for particular terminal arrangements, such as Zi = Zj = o,
the roots admit of rather easy determination.

The chief utility of the Expansion Theorem will be seen to reside in
the fact that by its use the solution for the transient oscillations of the
system is reduced to formulae which are functionally the same as those
for steady state oscillations, so that the problem is always completely
solvable provided the roots of the characteristic D{p) admit of deter-
mination.

November 15, 1916.

226 • HARRY NYQUIST. flSS?

THE STARK EFFECT IN HELIUM AND NEON.

By Hakry Nyquist.

THE effect of an electric field on spectral lines has been studied by
Stark^ in a number of substances but particularly in hydrogen,
helium and lithium. Lo Surdo* studied the effect by a different method
from that employed by Stark. Sonaglia' using Lo Surdo's method ex-
tended the investigations of hydrogen to the line Jf,. Koch,* using the
method of Stark, investigated helium and discovered two new lines
produced by the electric field. He also extended some of the results
previously obtained by Stark. Brunetti* applied Lo Surdo's method to
helium, but unfortunately it has not been possible to obtain a copy of his
paper. Evans and Croxson,* also using Lo Surdo's method, investigated
a mixture of helium and hydrogen particularly with reference to the
bearing of Epstein's theory on the line 4686.

In the present investigations discharge tubes both of Stark's and
Lo Surdo's type were employed and compared in preliminary experi-
ments. Stark's tube has the disadvantages that the intensity of the
light, from it is low and that it is difficult to replace when broken, while
the disadvantages of Lo Surdo's tube are that it breaks readily at the
cathode due to heating and that the glass about the Crookes dark space
soon becomes opaque owing to sputtering from the cathode.

An ideal discharge tube for observing the effect would be one that would
produce a great light intensity in a strong electric field without covering
the walls with opaque matter or otherwise changing with time. Un-
fortunately an improvement in any one of these directions seems to be
disadvantageous in others, so a tube must be something in the nature of a
compromise.

After much experimenting the form of tube illustrated in Fig. i was
adopted as being more satisfactory than any other type tried.

^ Elektrische Spektralanalyse chemiscber Atome, Hirzel. Leipzig* 19 14; Ann. d. Phjrsik,

48. P- 193. 1915.

> Accad. Lincei. Atti 32, p. 664, 1913; 33, p. 83, 1914.

* Accad Lincei. Atti 34. p. 631, 1915; N. Cimento, 11. p. 307. 1916.

^Ann; d. Physik, 48, p. 98, 1915. Cf. Stark, Elektrische Spektralanalyse chemiacher
Atome, p. 73.

> N. Cimento, 10, p. 34. 191 5*

• Phil. Mag.. 33, p. 327, 1916.

tS^Ci ^"^ STARK EFFECT IN HELIUM AND NEON. 227

The tube consists of a main portion M, of about 12 mm. internal di-
ameter, into which a bottle-shaped portion B is fitted rather loosely and
made tight with sealing wax, so it can be removed and inserted without
difficulty. Within the portion 5 is a solid aluminium rod of about 4.8
mm. diameter which serves as cathode. It is inserted so that one end,
which is filed flat, comes nearly flush with the narrow upper end of B.
A small vacant space is left between the sides of the
cathode and the glass to prevent conduction over the
glass. Resting on the upper end of portion B and
partly surrounding it is the aluminium cylinder D.
One half of this cylinder is bored to a diameter suffi-
cient to fit loosely over B. The remainder is bored
to a diameter 3.25 mm. The upper end of the larger
hole does not form a square shoulder but is slightly
curved. Thus the metal cylinder is not in contact
with the cathode but is insulated from it by means of
the glass. From a point opposite the upper end of
the cathode and extending about 7 mm. toward the
anode there is a slit S, .75 mm. in width, through the
wall of the aluminium cylinder. Opposite this slit
there is a side tube G whose outer end is covered by a
window tr. The anode ^ is situated about 12 cm. from
the top of the aluminium cylinder and at this place
there is a side tube T for exhausting the tube and in-
troducing the gas under investigation. The whole dis- Fig. 1.
chai^ tube is made of Pyrex glass. This glass has a
small coefficient of expansion and softens only at a high temperature,
both of which qualities make it a desirable material for the tube. The
electrical connections are made by means of platinum wires. These are
sealed through the glass without any special precaution and without the
use of any other kind of glass. The seal thus formed is not quite gas-
tight but can be made so by a small drop of sealing wax. Wax joints are
used to connect the window to the side tube (T, the side tube T to the
remainder of the apparatus, and part B to part M. As this last place is
heated by the discharge, two fine jets of air are employed to keep its
temperature down.

The spectrum was investigated by means of a spectrograph consisting
of six prisms made by KrUss and reground by Brashear. The faces of
the prisms are 6 cm. by 6 cm. The collimator lens is an achromatic
triplet whose focal length is 90.5 cm. and whose diameter is 6.5 cm. The
camera lens is a doublet whose focal length is 1 16 cm. and whose diameter

228 HARRY NYQUIST. [to»

IS 8.5 cm. These lenses, both of very good quality, are the property of
Professor Hastings, to whom I am indebted for kindly allowing me to
use them.

About 12 cm. in front of the discharge tube is a double image prism.
An achromatic photographic lens focuses the light that has passed through
the double image prism on the slit. The two images thus formed are
plane polarized, one parallel and one perpendicular to the discharge tube
and the slit. As was to be expected with so many prisms, it was found
that light having the electric vector parallel to the slit is reflected from
the faces to such an extent that it is almost impossible to photograph the
lines. To obviate this a mica half-wave plate having its axes at 45**
with the slit is placed in the path of that beam near the slit. This changes
the polarization so that the light from both images has its electric vector
perpendicular to the slit.

For some of the work in the red portion of the spectrum a plane grating
was used in place of the prisms and the mica plate was eliminated.
The grating is 8.0 by 5.3 cm., has about 15,000 lines to the inch and has a
total of 44,100 lines. The second order was used. The plane of the
grating is nearly perpendicular to the axis of the camera. The grating
being thus inclined to the collimater axis the full aperture is utilized.

The dispersion of the prism spectrograph varied from 2.1 A per mm. in
the violet to 8.2 A per mm. in the red. Both these figures are for mini-
mum deviation. On any one plate the light of shorter wave-length
suffers a greater dispersion and that of a longer wave-length suffers less
dispersion. The dispersion of the grating is 7.3 A per mm. in the second
order.

The electrical arrangement is indicated schematically in Fig. 2. This
is essentially the same kind of apparatus that is employed by Dr. A. W.
Hull,^ of the General Electric Company, for energizing X-ray bulbs.
His apparatus, however, supplies about ten times the voltage of the
present one. The present apparatus was obtained from the General
Electric Research Laboratory through the kindness of Drs. Whitney and
Hull.

The source of energy is a i lo-volt alternating current circuit. Tt and
T% are transformers which step down the voltage to about 25 volts.
The primary current is regulated by means of the variable resistances
ft and ft. K and K' are hot-wire rectifiers or kenotrons, the filaments
of which are kept incandescent by the current from the transformers
Tt and Tj'. The anodes of the rectifiers are connected together. Ti is a
transformer which steps up the voltage from no to 13,200 volts. The

> Gen. El. Rev., 19, p. 173, 1916.

VOL.X.1

Naa. J

THE STARK EFFECT IN HELIUM AND NEON,

229

LJKH

Fig. 2.

middle point of the secondary is grounded to the transformer case and
is connected to one side of the capacity C, the other side of which is con-
nected to the anodes of the rectifier. The two ends of the secondary
of transformer Ti are connected
to the middle points of the sec-
ondaries of transformers Ti and
Tt respectively. The seconda-
ries of the transformers Tt and
Tt are in turn connected to the
filament of the rectifiers. The
capacity (.17 microfarad) con-
sists of 100 small commercial
condensers connected in parallel.
The condensers are kept continu-
ously charged by the rectifiers and are continuously discharging through
the inductance L, "the ballast resistance R and the discharge tube D.

The inductance L consists first of a coil of carrying capacity of 50
milliamperes and with an inductance of 400 henrys. To this was added
the secondary of an induction coil of unknown inductance. The ballast
R consists of a rectangular sheet of asbestos painted on one side with
lampblack and wood alcohol and has a resistance of about .8 megohm.
The discharge tube D is shown in detail in Fig. i.

The tube was exhausted by means of a Geissler pump. A reservoir
containing the gases was connected to the tube by means of sto[)cocks
in such a manner that a small portion could be admitted at each turn
of the stopcocks. Another Geissler pump served to transfer the gas from
the tube back into the reservoir, when not in use. A charcoal bulb and
a U-tube were connected to the tube as near to it as convenient. They
were immersed in liquid air and served to withdraw all gases and vapors
from the tube with the exception of hydrogen, helium, and neon. The
pressure was measured by a McLeod gauge. A palladium tube was
attached to the apparatus and served to introduce hydrogen by being
heated in a hydrogen flame.

The helium was prepared by Professor Boltwood from thorianite.
The neon was produced from crude argon by freezing out the other con-
stituents with charcoal and liquid air. The crude argon had been pre-
pared by Professor Boltwood from atmospheric air by passing it through
a mixture of CaCa (90 per cent.) and CaCla (10 per cent.) heated to bright
redness. The neon contained appreciable quantities of helium and
hydrogen. My thanks are due to Professor Boltwood for kindly putting
the gases at my disposal and for assisting in their manipulation.

230 HARRY NYQUIST. [SSS

For wave-lengths less than 5,000 A., Seed 30 plates were used, while
Cramer's Spectrum plates were employed in the red and yellow. For a
short region in the green neither of these plates was very satisfactory and
in this region the Seed 30 plates were used after being stained with ery-
throsin. The recipe used is that given in Baly's Spectroscopy, p. 351,
1st ed. The plates were cut into strips 25 cm. by 2.5 cm.

The range of pressure which is suitable is rather small. If the pressure
is increased above ordinary working conditions, the conductance of the
tube is very much increased, the drop of potential and hence the field
becomes too small for satisfactory work. On the other hand, if the
pressure is decreased much the luminosity of the discharge diminishes
rapidly and soon ceases altogether. The best working pressures with
the present apparatus were found to be for helium about 2.6 mm. of
mercury and for neon about 1.5 mm.

The fall of potential across the tube as measured with an electrostatic
voltmeter varied rapidly with small changes in the* pressure. Under
working conditions the fall of potential was 4,000-6,000 volts. The
current varied from 2 to 8 milliamperes.

The times of exposure varied from 2 min. to 13 hrs. depending upon
the region of the spectrum investigated and the intensity of the lines.

When the current is turned on, the positive rays in the region above the
cathode collect at the axis of the aluminium cylinder where they form a
narrow but very luminous beam. This beam is the source of light and
being situated in the cathode fall of potential is affected by a strong field.
The beam or stream of positive rays rapidly attacks the aluminium
cathode when freshly prepared and digs a pit in its center. While this
pit is forming the electrical field is not stable nor is the discharge even and
continuous. After having run for about an hour, however, a stable con-
dition appears. The field stays constant and the discharge appears to be
continuous. The pit is then conical, has a diameter of about .5 mm. and
a depth of about 1.5 mm. and changes only very slowly.

After being run for about 40 hrs. a point is reached when a black film
accumulates about the cathode on the glass surrounding it and in the
aluminium cylinder. Then a condition of instability again sets in and it
becomes necessary to take the tube apart at the wax joint and clean
the parts. At no time is there any trouble from sputtering on the
window. This is a great advantage over the original Lo Surdo tube where
the glass is close to the cathode.

While no attempt was made to study the Balmer series of hydrogen,
photographs of these lines were obtained incidentally. They serve as a
comparison of the resolution of the present apparatus with apparatus

No!"i^*] THE STARK EFFECT IN HELIUM AND NEON. 23 1

used formerly. All the strong components given by Stark for H^, Hy,
and H^ appear. The weak components given by Stark do not appear,
probably because with the amount of hydrogen present the exposure
was not long enough. It is possible, however, that their relative inten-
sity is greater in the strong fields employed by Stark. The moderately
strong components do appear with the exception of the outer ones vi-
brating parallel to the field in H^ and H^. For H. the resolution is
essentially that attained in Stark's '* Grobzerlegung " or rough analysis.
The number of components is in each case except fl. greater than hitherto
obtained with the ordinary Lo Surdo tube.

If the frequency of a spectral line is affected by the field, it follows
that its image on the photographic plate is no longer a straight line in
its usual position but its various points are displaced, the displacement
being a function of the field strength. Moreover, since the field is a
continuous function of the distance from the cathode, the line on the
plate will in general be changed into one or more curved lines. With the
particular construction employed the field strength has a maximum at the
lower end of the narrow hole in the aluminium cylinder, which point is
situated about .5 mm. above the cathode. The field falls off in both
directions from this point and has three-fourths its maximum value at
the cathode and reaches a value very nearly zero at a point about 4 mm.
above the cathode, this distance being a function of the pressure. It
should be pointed out in this connection that the field depends on the
diameter of the hole in the aluminium cylinder; the smaller the diameter
the greater the field. It is possible to increase the field by making the
hole narrower, but the intensity is decreased as a result.

Stark's measurement of the effect in the lines of the Balmer series of
hydrogen was carried out with considerable precision, and, as the hydro-
gen lines were present along with those of helium and neon, his measure-
ments have been used in the present investigation for determining the
field. An absolute method would be to integrate the displacement along
the line, i, «., to find the area inclosed between the original line and the
displaced line. Then this area would be to the total drop as the dis-
placement at any point is to the field at that point, assuming that the
displacement is proportional to the field. However, this method would
involve a separate series of exposures to establish such linearity of
relation, and the accuracy would probably be much less than that ob-
tained by reference to Stark's results in hydrogen. It should be stated
here that the potential of the aluminium cylinder is nearly that of the
anode, a preliminary experiment showing a difference of less than 200
volts between them.

232 BAMRY NYQUIST.

The relation between the dis|dacement and the field is in general
expieased by the relation

SX^ a + bE + cP + etc.,

where SX is the dj^>lacenient of a given component, E the electric field
intensity, and a, 6, c, etc, are coefficients independent of £. The
measurements of Stark show that all these coefficients with the exception
of b are zero for all the components of the lines of the Balmer series of
hydrogen. The idea that a may be different from zero appears strange
at first consideration, since it signifies a definite displacement for zero
field. It might be argued that if there is a displacement for zero field,
we should see in the spectrum from an ordinary discharge tube, not single
lines but groups of lines, doublets, triplets, etc. The explanation of this
apparent contradiction to observed facts is that the intensity of a given
component as well as its displacement is a function of the field strength.
In components where a differs from zero, the light intensity approaches
zero as the field strength approaches zero. The method of finding a
will be understood from Fig. 3 (c), which illustrates four such components.
It will be seen from that figure that these components approach asymp-
totically a line which is parallel to the undisplaced li^e. The distance
between these parallel lines is a measure of a. The presence of terms in
£*, etc., is investigated by comparing the components of the lines with
the components of the Balmer series of hydrogen. If c has an appreciable
value for any component of a helium or neon line, its form will differ
from that of the components of the hydrogen lines. No such difference
has been found. Hence it will be assumed that with the field strength
employed cE^ is negligible and that

ax = a + bE.

Components in which a differs from zero are found mainly in the helium
lines, but a few are also found in some neon lines. Such components may
be looked upon as new lines, especially as a number of new lines appear
which are not components of any known lines, but since it is obvious that
they are closely related to certain undisplaced lines, it is perhaps best to
treat them as components.

In some of the earlier plates a comparison spectnmi was used produced
by letting light from an ordinary capillary discharge tube fall on the slit.
The comparison spectrum consisted of three sections; one between the
two spectra under investigation, one above, and one below them. By
thus having three sections it was possible to eliminate the uncertainty
arising from the curvature of the lines, which curvature is inevitable in
prism spectrographs. It was found that there was no displacement

No'a^*] ^^^ STARK EFFECT IN HEUUM AND NEON. 233

between the lines in the comparison spectrum and the upper part of the
lines investigated, hence, it was assumed that the field is zero at the source
of such upper portion. This assumption is further borne out by the fact
that lines which are symmetrically divided by a field and which therefore
would be broadened if a field was present show no such broadening.

In Fig. 3 are shown some illustrations of the lines as they occur on the
photographic plate. In these drawings the wave-length increases from
left to right. The upper part of the illustrations show the line as it ap-
pears when the field is zero; the lower part shows the effect of the field.
The pair of components illustrated at (a) is typical of the Balmer series
All the components in that series, except the central unaffected ones,
seem to arrange themselves in such
pairs. In helium there is only one line
(4,686) which has such a symmetric pair

of components and in neon there are /^ ^ I (T f\^ (
none. The form shown in (b) is typical
of lines which show no appreciable Stark
effect. While it is distinctly broadened

in its lower portion there is no doubt that ^ ^^ // / \i 4

at least the greater part of that broaden- ' // 11 Iv Jj ■

ing is due to increased intensity in the Fig. 3.

stronger field. The illustration (c) rep-
resents those components of He 4,388 whose electric vector is per-
pendicular to the field. It shows in order from left to right two com-
ponents having a < o and 6 < o, one component having a < o and
6 = 0, one having a < o and 6 > o, and finally two having a = o and
6 > o. The line He 4,922, sketched at (d), illustrates a line having one
of its components so far removed that it might well be looked on as a new
line. However, the intensity and general appearance of this component
indicates that it is closely related to the other components of the line.
The type (e) is very common in neon as is also type (f). It is possible
that some of the lines found to be of the type illustrated at (e) are in fact
of the type illustrated at (/), the resolving power of the spectrograph being
too small to separate them. A few lines in neon are of the general type
illustrated at (g). Unfortunately the components farthest away are in
this case so faint that they are difficult to measure. A number of lines
of the type shown at (A) occur, particularly in neon. They will be called
for convenience new lines. They seem to appear only in the electric
field where they are very broad and intense, several times as intense, in
fact, as any other neon line in the same region. Where the field is zero
these lines disappear altogether, or at least become so faint that they leave
no impression on the photographic plate.

234 HARRY NYQUIST. \^Sm

Helium.

The results obtained in helium are tabulated in Table I. Each line in
the table refers to a component. The first column indicates the wave-
length of the line unaffected by the electric field. The second column
gives the field strength, and the third indicates the means whereby E
has been calculated, a indicates that E has been computed from the
distance between the parallel components of H. and the data given for
that line by Stark.^ Similarly P indicates that the field has been obtained
from the data given for the parallel components 5— 5 of H^*. Also 7
refers in the same manner to components 6 —6 of H/ and 5 to components
7 — 7 of H3*. In the next column, headed polarization, is indicated
whether the component has its electric vector parallel to the field (p) or
perpendicular to the field (s) or whether there is a component in both
images (ps). The next column gives the change in wave-length due to
the field. A positive value indicates that the wave-length is increased, a
negative one that it is decreased. The last two columns give the coef-
ficients in the relation 5X = a + bE. The units are A and volts per centi-
meter throughout.

The displacement was found in most cases by measuring the distance
on the plate between hydrogen lines of known wave-lengths, one on each
side of the line under investigation and assuming that the distances are
proportional to differences in wave-length within this region. When
no such lines of reference exist close enough together, the table given by
Merwin* has been employed after being tested on known lines.

The following new lines were reported by Koch: 4,519, 4,046. In ad-
dition the following new lines appear on my plates: 3,962, 3,946.

In referring to the components, the following convention will be em-
ployed to identify them. First, the wave-length of the undisplaced line
will be given, then, in order, the numerical values of a and of b, and finally,
if necessary, the letter ^, 5, or both.

The component 4,686, o, o, is probably made up of several components
but the present apparatus does not separate them. Most of the other
components in helium are very sharp and are probably not further separ-
able. The components 4,472, o, o, ps, are weaker in the stronger portions
of the field, the contrary being the general rule. The component 4,388,
— .80, —1. 16, p also is weakened as the field increases and is nearly in-
visible at the point of maximum field, whereas the component 4,388,

^ Elektrische Spektralanalyse chemischer Atome, p. 51.

* Loc. cit., p. 54.

* Loc. cit., p. 55.

* Loc. cit.. p. 56.

* Am. Jour. Sci., 43, p. 49, 1917.

VOL.X.1

No. 3. J

THE STARK EFFECT IN HELIUM AND NEON.

235

Table I.

Helium.

£.

Computed
Prom .

Polarisation.

ax.

6678

30,900

Doubtful

5876

5047

20,000

5015

4922

2.31

1. 16X10-*

1.49

.75

-2.31

-1.24

-.54

-11.90

-11.35

-.28

4713

38,600

Doubtful

4686

1.24

.32

-1.24

-.32

4472

36,400

1.17

.32

.78

.20

-3.12

-1.52

-.43

-3.80

-1.52

-.63

4438

.58

.16

4388

6.33

1.74

5.75

1.58

5.50

1.51

3.67

1.01

.86

-.40

.13

.61

-.40

.06

-.40

-5.02

-.80

-1.16

-8.58

-3.61

-1.36

-9.64

-3.61

-1.66

4169

26,200

1.12

.43

4144

6.90

2.64

5.17

1.97

2.97

1.13

1.57

.60

-.93

-.41

-.20

-2.17

-.41

-.67

-6.00

-.41

-2.13

-8.13

-.41

-2.95

-9.06

-.41

-3.30

4121

.09

-.03

4026

26.800

2.91

1.09

2.30

.86

-.54

-.70

-.54

-.06

-3.16

-1.08

-.78

-3.78

-1.08

-1.01

3965

-.44

-.16

-.73

-.27

3889

Doubtful

236 HARRY NYQUJST. [I

— .80, —1. 16, 5 is Strengthened with an increase of the field. The true
explanation probably is that the component in question 4,388, —.80,

— 1. 16, ^5 is elliptically polarized and that the eccentricity of the ellipse
increases with the field. For any line having several values of a those
values are simple multiples of the least one.

An examination of the table discloses the fact that the lines may be
divided more or less sharply into types and that the lines of any given
series are in general of the same type.

The line of the principal series of helium (3,889) does not show any Stark
effect or, if it does, it is too small to measure. In the first subordinate
helium series three members are represented: 5,876, 4,472, 4,026. They
show a progressive change as follows. The first line has one group of
components (probably only one component) ; the second has two groups
with different values of a and the third has three such groups. The
second subordinate helium series shows a small effect in the two lines
representing it in the table (4,713, 4,121). In both cases the effect
seems to be nearly the same, merely a single component displaced slightly
toward the red. The lines of the first subordinate series of parhelium
(6,678, 4,922, 4,388, 4,144) are separated into more components and the
components <ire farther separated than in any other series. The lines of
this series resemble each other very much on the photographic plate.
The second subordinate series of parhelium (5,048, 4,438, 4,169) resembles
the corresponding series in helium in that its members are composed of a
single component having a positive 5X. The values of 5X are greater
than in the helium principal series. The line 3,965, which is the only
member of the parhelium principal series showing a measurable deflection,
is exceptional in that its component has a =» o, 6 < o. No other line in
helium or neon shows this effect. The line 4,686 is very much like the
line Ha- It is in fact different from any other helium line. This line is
interesting from the point of view of Epstein's theory* of the Stark effect.
It has unfortunately not been possible to obtain a copy of Epstein's
paper, but according to Evans and Croxson* it demands the value 24/7
= 3.43 for the ratio of the separation of H^ to that of 4,686. The actual
ratio found is 4.75 (nearly 24/5) a discrepancy of about 38 per cent. In
other words the ratio instead of being 24/(4^—3*) is 24/(3'— 2').

Neon.

As has already been pointed out a very general phenomenon is the
increase of the intensity in the portion of the lines which corresponds to
the field. It seems likely that this is not due to the field directly but may

> Epstein, Phys. Zeitschr., 17, 148, 1916; Ann. d. Physik, 50 ($), 489, 1916.
■ Loc. cit.

VOL.X.1

THE STARK EFFECT IN HELIUM AND NEON.

237

Table II.

Neon.

£.

dA.

6206

30,900a

.I7u

.05X10-*

6189

ASu

.04

6175

.80

.26

6151

.SOu

.16

5992

.43

.14

5988

.53

.17

5976

.70

.23

5966

.57

.18

5962

.30fi

.10

5919

.33

.11

5914

.50

.16

5907

1.20

.39

5903

.60

.19

5873

.35

.11

5820

.57

.18

5812

22,500n

(blurred by an H line)

5805

30,900a

.99

.32

5765

.62

.20

5761

22,500n

.15

.07

5748

30,900a

.74

.24

5719

22,500»

.27

.12

5690

.18

.08

5657

.27

.12

5653

.32fi

.14

5589

.34fli

.15

5563

.45

.20

5419

20,000^

1.07

.54

5413

1.43

.72

5383

1.43tt

.72

5375

3.03

1.52

5356

3.21

1.61

5333

2.97

1.49

5327

.40

.20

5214

1.19

.60

5211

1.38

.69

5209

1.80

.90

5204

2.76

1.38

5193

1.60

.80

5189

29,600n

.13

.04

5159

20,0009

1.78

.89

5155

2.15

1.08

5152

29,600n

4.18

1.41

5117

.51

.17

5114

20,0009

.26

.13

4945

29,600»

.71

.24

4939

.32

.11

4892

38,600^

1.00

.26

4866

9.00

2.30

238

HARRY NYQVIST.

[

Table II. — Continued.

£.

5A.

4822

38,60Q|9

.44

.11X10-*

4819

3.54«

.92

4790

4.65tt

1.20

4789

.33

.09

4750

6.96«

1.80

4713

2.12tt

.55

4712

7.96

2.06

4709

1.68

.44

4703

2.48

.64

4646

.14

.04

4615

36,3007

1.25

.34

4583

1.13

.31

4575

5.45

1.50

4425

6.12

1.69

4423

3.30

.91

be due to other causes such as more complete ionization. At any rate
the effect of this increase in intensity is to broaden the line on the photo-
graphic plate. Now if the displacement of such a line is small it may well
happen that the broadening masks the displacement either completely
or to such an extent that the displacement can not be measured.

The following neon lines showed no displacement but were broadened.
The ones marked with an asterisk were investigated by means of the
grating as well as with the prisms. The field strength as computed from
Stark's data for H. was 30,900 volts/cm.

7»059. 7.033. 6,930, 6,717*, 6,678*, 6,599*, 6,533*, 6,507*, 6,445, 6,410,
6,402*, 6,383*, 6,352, 6,335*, 6,331. 6,328*, 6,314, 6,305*, 6,294, 6,267*,
6,247, 6,217*, 6,214, 6,182, 6,164*, 6,143,* 6,129, 6,118, 6,096*, 6,074*,
6,046, 6,030*, 5.975, 5.945*. 5.939. 5.882*, 5,852*, 5,829, 5,663, 5434,

5.401. 5.372. 5.234. 5.189.

The following neon lines show a positive displacement, but it is so
small compared with the broadening of the line that it can not be meas-
ured. The field is from 20,000 to 30,900 volts/cm.

6,001, 5,349, 5.343. 5.341. 5.331. 5.320, 5,305, 5,298, 5,280, 5,274,
5,222, 5,150, 4,837, 4,828.

The lines in Table II. have one component parallel and one perpen-
dicular to the field. Further the two components appear to be displaced
equally, which makes it probable that the light is unpolarized. The
letter (a, j8, 7, n) after the field strength refers to the known line from
which the field has been computed. In this connection the letter n refers
to the neon line 5,204, which was used in some cases, and for which the
constant has in turn been computed from H^. The value of a is zero

NoI"3^] ^^^ STARK EFFECT IN HELIUM AND NEON. 239

for all lines' in this table. The letter u after a number indicates that,
by reason of obscurity of lines or other causes, the measurement is un-
certain.

The component of 5,117 is probably double. The line 4,713 is blurred
by the helium line.

In Table III. are listed the neon lines which have more than one
component. The notation is the same as in Table I.

The component 5,360, o, 1.25, ps, is very faint in comparison with
the components 5,360, o, o, ps. The component 5,074, — 4.30, — 1.31,
5, is faint and blurred. It may consist of two components. The com-
ponent 5,074, — 4.30, — 1. 31, 5, is so weak on the plate that its presence
can not be established with certainty. The component 5,038, 0.96, ps,
may be made up of two. The components 5,031, — 4.75, — 1.69, p,
and 5,031, — 4.09, — 1. 14, pf are uncertain. The component 4,810,
— 4.65, —.52, 5, probably consists of two components.

A considerable number of new lines appear. These are very much
more intense in the field than any other neon line in the same region, but
the plates show no trace of them where the field is zero. The field
strengths given in connection with these lines is the maximum field in
the tube at the time of exposure. They do not indicate that those field
strengths are the minimum required to produce the lines.

A field of 20,000 (j8) produced the new lines 5,200, 5,188, 5,149, 5,139,
5,073, and 5,071. A field of 36,400(7) produced the new lines, 4,616,

4.589» 4.569. 4.556, 4,555, 4,534. 4.533. 4.5^4. 4.513. 4.500, 4.458. 4,430,
4,427, 4,420, 4,413, 4,412, 4,409, 4,402, and 4,392. A field of 34,800
produced the lines 4,380, 4,371, 4,307, 4,291, 4,253, 4,242, 4,235, 4,230,
4,228, and 4,216.

The new line 5,139 has a displacement 5X = .20 or 6 = .10 X io~*.
The line 5,071 may be the hydrogen line.

These tables and lists probably are not complete even in the region of
the spectrum which they cover (the visible). Some relatively strong
lines are split up into components some of which are so faint as to be
barely detectable. It is thus quite possible that other fainter lines have
components that are too faint to be detected with the present means.
This will be appreciated when the vast number of faint neon lines is taken
into consideration. Further a number of known faint lines do not appear.

On examining the data given above, certain general facts are evident.
They may be briefly summarized into the following rules which are
applicable to helium and neon only.

I. The Stark effect increases with the frequency and more rapidly than
the first power of the frequency.

240

HARRY NYQUJST.

Table III.

Neon.

£.

Polarixation.

«x.

5360

20,000/9

2.50

1.25 X 10-*

5145

29,600»

3.46

1.17

2.24

.76

5122

3.52

1.18

2.38

.80

5081

2.76

.93

1.65

.56

5074

4.23

1.43

- .31

-1.48

.40

-2.70

-3.27

.19

-8.18

-4.30«

-1.31

5038

2.85

.96

1.50

.51

5031

3.89

1.31

- .16

-1.38

.41

-2.59

-3.06

.16

-7.47

-4.09

-1.14

-9.75

-4.75

-1.69

5005

3.17

1.06

1.90

.64

4957

3.28

1.11

2.03

.69

4885

38,600/9

4.15

1.08

1.55

.40

.28

.07

4863

9.12

2.37

6.95

1.80

4818

4.65

1.20

2.15

.56

-4.87

-3.92

- .25

4810

6.75

1.75

.33

-1.54

.47

-6.64

-4.65

- .52

4753

6.77

1.75

5.42

1.40

-2.33

.60

-6.96

-5.21

- .45

4715

7.85

2.04

5.70

1.48

4710

1.77

.46

-1.77

-0.60

- .30

4705

2.48

.64

-3.54

-4.65

.29

-4.86

-5.77

.24

-11.05W

-7.43

- .94

4541

36,400t

12.65

3.48

9.78

2.69

VOL.X.1

No. 3. J

THE STARK EFFECT IN HELIUM AND NEON,

241

Table III. — Continued.

£.

Polarisation.

ax.

4538

36,40(^7

9.05

iMxicn

(«

6.78

1.86

3.06

.84

1.69

.46

4488

4.70

1.29

4.48

1.23

.61

.17

.24

.07

4467

1.12

-31

.56

.15

2. Of two lines in the same region of the spectrum the weaker is usually
affected more than the stronger.

These two rules have frequent exceptions. Indeed, they are sometimes
contradictory, namely, when the line of greater intensity also has the
greater frequency. If we combine the two rules into one giving proper
weight to the two factors, the number of exceptions is small.

3. When a = o,bis positive. Only two exceptions have been found to
this rule, viz., He 4,686, and He 3,965.

4. a is never positive.

5. If a given helium line has several values of a these values are simple
multiples of the least one.

6. Where a = o the ratios of b for different components approximate
to simple numerical ratios. Sometimes this approximation is poor and
the discrepency is greater than the error in measuring.

7. When a = o for a group of components the s components are never
farther displaced than the p components; the p components are displaced
as far or farther than the ^ components. When a = o no corresponding
rule can be stated because as has been said the p components are then
too faint to be observed with certainty.

It was pointed out in discussing the helium spectrum that the lines
which belong to the same series are similarly affected. We should there-
fore expect a similarity in neon between lines of the same series. Some
neon series have been given by Rossi.* Unfortunately most of the lines
in these series are so faint that it has not been possible to get them on
the negatives. However, as far as the present data go some corre-
spondence with Rossi's series is suggested. The first series of Rossi is
represented by the following lines on my plates: 5,820, 5,765, 5,081,
51O38, 4,753, 4,715. The first two are given in the table as having one
component each. The lines 5,081, 5,038, and 4,715 show two components

> Phil. Mag., 26, 981, 1913.

242 HARRY NYQUIST. \^am.

each. 4,753 has more than two components and differs in this respect
from the other lines of the series but these components are faint and there
may be components corresponding to them in the other lines which are
too faint to make an impression on the plate. It is possible also that
5,820 and 5,765 may have two components and that the spectrograph
does not resolve them. The second series of Rossi is represented by the
lines 5,805, 5,748. 5,074, 5,031, 4,750, and 4,712. The lines 5,074 and,
5,031 differ from nearly all other lines in the number of detached com-
ponents. We should then expect such components in the other four lines
as well, and the plates have been carefully examined with this in mind.
As for the first two there are a number of hydrogen lines where the faint
components might be expected and nothing definite can be said about the
absence or presence of detached components. As for the pair 4,750 and
4,712 the plate does indeed show detached components in this region but
they have been attributed to other lines in the table. When several
neon lines are close together there is of course no certain way of telling
whether a detached component belongs to one line or another. If we
attribute to 4,750 the detached components which have been attributed
to 4,753 and to 4,712 the ones which have been attributed to 4,710 and
4»705, we not only increase the agreement between lines of the second
series but secure nearly perfect agreement in the first series. Moreover,
the lines in the second series of Rossi and the lines 4,818 and 4,810 will
then be the only lines in the neon spectrum which have detached com-
ponents.

Summary.

1. The Stark effect in helium and neon has been investigated by means
of a high dispersion prism spectrograph and a new type of tube which is
essentially a modification of the Lo Surdo tube.

2. It has been found that the various lines investigated may be classi-
fied in several types, and that lines which belong to the same series are
of the same type.

3. The components obtained have been tabulated.

4. It has been found that the displacement is approximately a linear
function of the field and that the absolute term in the equation relating
the displacement and the field is not always zero.

5. In the helium spectrum the two new lines produced by the field
and discovered by Koch were observed and, in addition, two other new
lines. In the neon spectrum thirty-four such lines are observed and
recorded.

6. A set of empirical rules has been given, which summarize quali-
tatively the results given in the tables.

/"/y. ^

Plate 1,

To face page 243

««*»** * ? I

^' ^v

k A

->^

ny. 6

HARRY

NYQUIST.

No!"3^*l ^^^ STARK EFFECT IN HELIUM AND NEON. 243

The present investigation has been conducted under the supervision of
Professor Bumstead, to whom I wish to express my thanks for constant
direction and encouragement. My thanks are also due to Professor
Taylor for assisting me in the more difficult glassblowing, particularly in
making a tube of the Stark type. My thanks are further due to Professor
Uhler for frequent advice about the use of the spectroscopic apparatus.

Sloanb Physical Laboratory,
Yalb Untversfty,
April 30, 191 7.

Dbscription of Platb I.

On Plate I. are illustrated portions of the spectrum that are of special interest. These
photographs were obtained with a mixture of neon, helium and hydrogen. As has been stated
previously separate photographs were taken with helium and hydrogen only.

While some of the lines are so faint as to be barely detectable, others are greatly overex-
posed. For this reason it was necessary to take several photographs of different times of
exposure. Those shown in the plate are of rather long exposure.

The upper spectrum in each figure is produced by light having its electric vector parallel
to the electric field, the lower by light vibrating perpendicularly to the field. These figures
all have the long wave-length end toward the left.

Fig. 4, exposure 3 hrs., voltage on tube 5,000, pressure z.5 mm., is from the blue portion
of the spectrum. On the left is shown the helium line 4,713 together with a number of neon
lines. To the right of this group are some detached components which have been mentioned
in the discussion of Rossi's series. An examination of the figure will make clear the difficulty
of assigning the detached components to the proper line. To the right of this group appears
the line 4,686, which, as has been pointed out, is the only helium line which shows a symmetric
effect. The helium line 4,472 is overexposed in this figure. The helium line 4,388 shows a
great number of components and has been illustrated in Fig. 3. In the present figure the
lower spectrum shows two images of this line. This is due to a fault in the optical system
probably in the double image prism. In most of the photographs this does not appear and
in some it occurs in the upper image. The line Hy shows the same defect, but in neither case
does it interfere with the measurements. On the negative, there appear a number of weak
components between the strong components in the upper image of Hy. These are nearly
lost in the process of printing. The small arrows below the figure indicate the new lines
which are situated in this region, one (4,519) belonging to helium, the rest to neon.

Fig. 5, exposure 5 hrs., voltage on tube 5,000, pressure 1.4 mm. This figure shows some
helium and some neon lines. Between the lines 5.038 and 5.015 several components appear.
They belong to the line Ne503i which is of the type shown at (g) in Fig. 3. The upper part
of this line appears on the negative but is practically lost in printing. Fig. 6, exposure 3
hrs., voltage on tube 6,000, pressure z.6 mm. This figure shows the line 4,92a in greater
detail than the drawing Fig. 3.

244 P' ^- BISHOP. [^J£

THE IONIZATION POTENTIAL OF ELECTRODES IN VARIOUS

GASES.

By F. M. Bishop.

THE object of the present investigation was to redetermine the ionizing
potential of certain gases under different experimental conditions
and to extend the work to some simple compounds and see if the com-
bination of one atom with another had any effect on its ionizing potential.

Two forms of apparatus have been used heretofore for the direct meas-
urement of ionization potentials; one in which the source of electrons was a
plate illuminated by ultra-violet light, was used by Lenard^ and by Dem-
ber,' and the other form in which electrons were liberated from a hot wire
or W^hnelt cathode was used by von Baeyer,* Franck and Hertz* and
Pawlow.* Since this work was in progress an important modification
of this second source has been described by Goucher.*

The results obtained by these observers with the two forms of apparatus
differ greatly, and it seemed desirable to find the reason for this dis-
crepancy by employing both methods of liberating the electrons and
using an apparatus in which the important quantities could be varied at
will.

The object in view in finding the ionizing potential of a simple com-
pound, the ionizing potential of whose components are known, was, first,
to determine whether the ionizing potential is an atomic property and
not dependent on the molecular combination; and second, if this first
proved to be the case, to open the possibility of determining the ionizing
potentials of some substances, which in their simple uncombined state
do not lend themselves easily to this direct method of determination.
Naturally the method would still be applicable only to substances in
which the unknown component ionized at the lower potential.

Apparatus and Method. — ^The method is essentially that described by
Franck and Hertz, Pawlow, and Goucher in which plate electrodes are
used.

1 p. Lenard, Ann d. Phys. (4), 8. 149, 190a.

« H. Dember, Ann d. Phys. (4), 30, 137. 1909.

» O. v. Baeyer. Verh d. D. Phys. Ges., 10, 96, 1908.

* Franck and Hertz. Deutsch. Phys. Ges.. Vol. 15. p. 34. I9i3-
» Pawlow, Proc. Roy. Soc., Vol. 90, p. 390, 1914.

• Goucher, Phys. Rev., Vol. 8, p. 561. 1916.

VOL.X.

THE IONIZATION POTENTIAL OP ELECTRONS.

245

The diagram of Fig. i shows the apparatus drawn to scale. i4 is a hot
wire source of electrons which could be readily changed, B a gauze to
which an accelerating potential is applied, and C a receiving electrode
made of oxidized brass which was found to be insensitive photo-electri-
cally. B is attached to a metal cylinder E and gauze screen F, which
completely enclose and shield the ionization chamber from any charges
that may accumulate on the glass walls of the containing vessel. G
and H are inlet and outlet tubes respectively through which a continuous
flow of gas is maintained during a set of readings by keeping a diffusion
pump running at one end and allowing the gas to diffuse through a small
capillary from a chamber in which the pressure could be suitably regulated
at will. On either side of the main apparatus were liquid air traps to
keep mercury vapor and also any vapor from the stopcock grease away
from the ionization chamber. Immediately beyond the liquid air trap
on the pump side was a McLeod gauge, and a discharge tube similar to
the one described by Pawlow in which the pressure could be suitably
regulated and the purity of the gas tested with a direct vision spectro-
scope.

The apparatus in which photo-electrons were used differed from the one
described in the following particulars, which are indicated in Fig. i b^

dotted lines. An aluminum plate was substituted for the hot wire A,
the apparatus contained a side tube L and quartz window ilf , and opening
N in the metal cylinder, so that a source of ultra-violet light could be
focused on the aluminum plate. The cylinder E could be moved by
means of an iron ring by a magnet placed outside the tube. A fine
copper wire in the form of a coil was substituted for the rigid contact K.
This tube contained a metal ring 0 inserted as shown in Fig. i to shield
the electrode C from scattered ultra-violet light as much as possible.
Ultra-violet light striking C, with the surrounding metal part £ at a
higher potential, will cause photo-electrons to be given off from C and
it will be noted that the resultant charging up of the electrometer is the

246 F. M. BISHOP. [^SS

same sign as that dae to the ionization we seek to measure. C was made
of oxidized brass to minimize this effect. Control readings were taken
with A and B at the potential of B and the positive deflections due to
this photo-electric effect subtracted from the ionization deflections. The
object in having the cylinder E movable was to be able to vary the
distance between A and B through which the electrons were accelerated,
and to set this distance less than the mean free path of an electron at
any pressure used.

Preliminary work in hydrogen made with this apparatus using ultra-
violet light showed that the form and position of the curve were not
affected by change of pressure, except when the distance AB was made
comparatively long and the pressure high enough so that additional
electrons would be given off due to ionization in the region AB and these
electrons in turn could acquire energy enough to ionize. This, it will be
observed, could not affect the form of the curve until the applied accel-
erating potential was twice the minimum potential required to ionize.

Since the relative distances had been shown to have no effect on the
ionizing potential by work with the first form of apparatus, the con-
struction of the second piece of apparatus, in which the hot cathode was
used, was considerably simplified. A rigid contact was substituted at
K for the flexible one previously employed and the iron ring was done
away with.

The distance between A and C in each case was 4 cm. and in the hot-
wire apparatus the distance between A and B was i cm. The cylinder
E was of brass and also the gauze £, the mesh of which was about i mm.

The writer wishes here to express his thanks to Mr. A. Greiner, of the
firm of Green and Bauer, who made both pieces of apparatus.

The electrometer used was of the Dolezalek type, sensitive to about
4,000 scale divisions per volt. This could also be used in connection with
a mica condenser which reduced the sensitiveness in the ratio of 9 to i.
The electrometer connection was shielded by an earthed metal screen P
insulated from the lead R by quartz insulators S and S\ Surface leaks
over the outside of the glass were prevented by an earthed metal foil T
moistened with a solution of calcium chloride.

During the process of obtaining an ionization curve the potential of A
was kept 4 or more volts higher than C, so no electrons from A could
reach C. The accelerating field AB was varied by varying the potential
on B by means of storage cells and a potentiometer. Since the retarding
field between B and C is greater than the accelerating field between A
and B no deflection of the electrometer will be observed until the electrons
acquire sufficient energy in the region ABto cause ionization in the region

vlS^i^'] ^^^ IONIZATION POTENTIAL OF ELECTRONS. 247

BC. When this potential is reached positive ions are driven to the
negative electrode C. This potential between A and B at which positive
ions begin to collect on C is taken as the minimum ionizing potential of
the gas. If the value of the ionizing potential is sought with extreme
accuracy corrections must be applied for two reasons as was pointed
out by Franck and Hertz and by Goucher. There is a drop in potential
of at least 6/10 volt between the two ends of the wire A due to the heating
current. Conduction of heat to the leads causes the ends of the wire to
assume a much lower temperature than the middle. Each of these
causes tends to make the velocities of the electrons ungual. For a very
accurate determination of the ionizing potential the equi-potential equi-
temperature source described by Goucher is undoubtedly superior. This
form of a source was not used in the present investigation because the
experiments were well under way before the method was published.

The accelerating potential recorded throughout this paper is the one
between B and the lower potential end of A. On account of the initial
velocity with which the electrons leave the hotter central portion of the
wire, this voltage more nearly represents the energy of a large fraction
of the electrons leaving A, and smaller corrections have to be applied
than if the voltages between B and the positive end of the wire were taken.

In order to determine the relative number of electrons coming off with
any particular velocity, electron current readings were taken where a
potential slightly below the ionizing potential was applied between A
and B and successive retarding potentials applied between B and C.
These electron currents were then plotted against the difference between
accelerating and retarding potentials and this curve was then graphically
differentiated, and the tangents plotted against the corresponding vol-
tages of the electron current curve. This gives a velocity distribution
curve in which the ordinates are proportional to the number of electrons
coming off with any particular velocity. After a number of electron
current curves had been taken at any given pressure it was observed that
these curves had their maximum slope approximately at zero volts; that
is, when the accelerating potential applied between A and B was equal
to the retarding potential applied between B and C. Further it was
observed that the position of maximum slope could be altered by changing
the heating current. In order to make the correction, which would
eventually have to be applied to the ionization curve, as small as possible
the heating current in the hot wire source of electrons was so regulated
by successive trials that the resulting electron current had its maximum
slope at zero volts. The value of this maximum tangent was set equal
to 100. The ordinates at other potentials on this curve represent the

248 p. M. BISHOP. I

number of electrons coming off at these potentials on the same arbitrary
scale.

Measurements in Hydrogen. — Hydrogen was prepared electrolytically,
dried by passing over calcium chloride and phosphorus pentoxide and
passed over heated copper in an electric furnace to free it from any trace
of oxygen. The hydrogen then passed through a trap immersed in liquid
air and into the ionization chamber.

The author wishes here to express his thanks to Professor Boltwood for
his suggestions and assistance in the preparation of the different gases
used.

With the photo-electric apparatus a break occurred in the curve for
hydrogen at about 16 volts as shown in Curve (g), Fig. 2, while with the
hot wire source this break occurred at 11 volts; Curve (a), Fig. 2, being
typical for this source. Since the electron current from the hot wire was
much larger than the current from the ultra-violet light source, readings
were taken with the current from the hot wire very much reduced, and
in this case curves similar to (g) could be reproduced. A very intense
ultra-violet light source, moreover, gave curves such as (/), Fig. 2,
where a break occurred below 16 volts but a sharp bend in the curve
occurred at about 16 volts. Curves (b) and (c) were taken with inter-
mediate electron currents. They show ionization beginning at 1 1 volts
and a sharp increase at about 15.7 volts. This fact shows that there is
a second and more intense type of ionization which begins at this higher
potential. Goucher and Davis, after being informed of this result, have
recently confirmed it with their apparatus.

The lack of complete agreement of this second break in the curve for
the two sources, 16 volts for one and 15.7 for the other, may be attributed
to the fact that a considerable correction has to be applied to the curves
(J) and (g). The electron current curves with the ultra-violet light source
did not have their maximum slope where the accelerating potential
between A and B was equal to the retarding potential between B and C,
but where the accelerating potential AB exceeded BC by several tenths
of a volt, that is, some of the applied energy was used up in helping the
electrons out of the plate, so that we may conclude that the two methods
are in good agreement and that this new second type of ionization in
hydrogen begins at about 15.7 volts.

The experiments in which a hot wire source of electrons was used give
1 1 volts for the ionizing potential of the first type of ionization in hydrogen
in agreement with the work of other observers. With the ultra-violet
light source of electrons it is shown that the number of electrons is usually
not sufficient to permit the ionizing potential of the first type to be
measured, and in some cases its presence may not even be detected.

Voi.X.1
No. 3. J

THE IONIZATION POTENTIAL OF ELECTRONS.

249

The pressures in the work on hydrogen varied between .001 mm. and
.03 mm., and the point at which ionization began seemed entirely inde-
pendent of the pressure within these limits.

Nitrogen. — Nitrogen was prepared by heating ammonium chloride and
sodiiun nitrite, bubbled through sodiiun hydroxide solution to remove
any carbon dioxide formed. It next passed over heated copper and
heated copper oxide in order to remove oxygen and hydrogen. It was
dried by passing through calcium chloride and phosphorus pentoxide,
then passed through a trap immersed in liquid air into the ionization

Fig. 2.

Fig. 3.

chamber. Curve (a), Fig. 3, is a typical curve for nitrogen, in which
ionization b^ns at about 7.5 volts. This curve was taken at a pressure
of .012 mm.

Oxygen. — Oxygen was prepared electrolytically and purified as de-
scribed for hydrogen, with copper oxide substituted for copper in the
electric furnace to remove any hydrogen present. Curve (c) is an
ionization curve for oxygen taken at a pressure of .0021 mm. The
apparatus had been previously used for nitrogen and while the main part
of the ionization starts at about 9 volts, traces of ionization begin before
this, due probably to small amounts of nitrogen given off from the
platinum strip, though the strip had been previously heated to only
slightly below its melting point for several hours with the diffusion pump
running continually.

There are certain difficulties connected with work in oxygen not ex-
perienced in other gases. When pressures of .01 mm. or above were used

250

p. M, BISHOP.

[Sbcond
r

no appreciable amount of ionization could be detected even several volts
above the ionizing potential of oxygen. This is no doubt the same effect
that Franck and Hertz attribute to charged double layers. By using
pressures of only a few thousandths of a millimeter this difficulty was
avoided. So the curves for oxygen were taken under these conditions.
Mercury Vapor. — Mercury vapor was introduced into the apparatus
by removing the freezing mixture from the traps and pumping down to
less than .ooooi mm. as indicated by the gauge, then allowing the mercury
vapor from the various mercury columns to diffuse back into the appa-
ratus.

Franck and Hertz and also Goucher obtained positive electrometer
deflections at 4.9 volts in mercury vapor, which they interpreted as
ionization.

No ionization could be detected in this experiment below about 10

volts even with electron currents
much more intense than those em-
ployed in any of the other gases.
For this purpose a tungsten wire
was used instead of the platinum
strip previously employed. Ioniza-
tion had definitely started at 10
volts, but a glance at the corre-
sponding velocity distribution curve
(&), Fig. 4, shows that a consider-
able correction has to be applied
to allow for the initial speed of
the electrons leaving the wire. For
example, the curve shows that the
number of electrons which have
a velocity corresponding to {AB
4- 0.5) volts is 1/5 as large as the
number having a velocity corresponding to AB volts. If this number
is sufficient to cause a measurable amount of ionization it would have
the effect of pushing the curve half a volt in the direction of greater
voltage. So we may safely conclude that the bend in the mercury curve
represents the 10.27 type of ionization and that these experiments
showed no ionization below this point. This higher value for mercury is
in agreement with the result recently obtained by Goucher and Davis
and presented at the New York meeting of the American Physical
Society on February 17. They showed experimentally that what had
been considered ionization occurring at 4.9 volts in mercury vapor was

Fig. 4.

No'i^*] ^^^ IONIZATION POTENTIAL OF ELECTRONS. 25 1

really photo-electric eflFect on the receiving electrode due to radiation
from the mercury vapor in the tube which was bombarded by electrons
having a velocity between 4.9 and 10.27 volts.

In the apparatus used in this experiment the receiving electrode was
of brass slightly oxidized. This was chosen for the ultra-violet light
apparatus because it had been tested and found to be very insensitive
photo-electrically. The same metal was used in the ho.t-wire apparatus.
This undoubtedly accounts for the fact that no positive deflection of the
electrometer occurred below 10.27 volts in mercury vapor as had been
found by other investigators. Hence it seems fair to assume, since the
present apparatus is not sensitive to photo-electric eflFect from radiations
in the tube, that the results obtained represent true ionization potentials
in the other gases also. If this is not the case then certainly the radiation
in these other gases is much more intense than in mercury vapor.

The ionization produced by electrons in mercury vapor is very much
more intense than it is in the other gases used, that is, after the ionizing
potential has been reached a much larger fraction of the collisions result
in ionization in mercury than in the other gases. This fact doubtless
explains the results obtained for ionizing potentials by Lenard, who found
the same value of 11 volts for all the gases tried. Since the mercury
vapor was not frozen out, and since it has been shown that ultra violet
light does not give a sufficient supply of electrons for the purpose, no
ionization was noticed in any case until the ionizing potential for mercury
was reached, which showed itself because of the relatively large number
of ions formed in this substance.

Nitrous Oxide, — ^The nitrous oxide used was taken from a cylinder of
nitrous oxide prepared and purified for medical purposes. A large
fraction was first allowed to escape from the cylinder and the gas used
was taken from the middle of the cylinder.

One of the objects of this experiment was to try a compound, the
ionizing potentials of both components of which had been measured
separately with the same apparatus, to see how these values would be
related to the ionizing potential of the compound. Nitrous oxide seemed
to lend itself naturally to this purpose.

The condensation point of this gas being — 92° C, a freezing mixture
of carbon dioxide snow and acetone having a temperature of — 78.2° C.
was substituted for the liquid air. Curve (&), Fig. 3, is an ionization
curve for nitrous oxide. It will be observed that this is almost an exact
duplicate of the curve for nitrogen which would indicate that the com-
bination of one atom with another in a compound had no effect on its
ionizing potential. Ionization was increasing too rapidly by the time

252 F. M. BISHOP. [i

the ionizing potential of oxygen was reached to detect any increase in
the curve at this point due to oxygen.

Owing to other plans, the investigation had to be terminated before
more compounds could be investigated.

The writer wishes here to express his thanks to Professor Zeleny for
his suggestion of the general field of investigation as well as to Professors
Bumstead and Taylor for their interest and assistance in the work.

Summary.

1. A comparison has been made of the two methods previously used
for determining the ionization potentials of gases by electrons, and the
method, where the electrons are liberated by ultra-violet light, is shown
to give misleading results because the number of electrons set free is too
small. This explains the discrepancy between the results hitherto
obtained by this method and those obtained with apparatus where the
source of electrons was a hot metal surface.

2. The ionizing potential of several gases has been determined under
conditions which tend to minimize the photo-electric effect on the re-
ceiving electrode due to radiations in the tube. Results were obtained
in good agreement with the accepted values for the following gases:
Oxygen 9, nitrogen 7.5, hydrogen 11.

3. In hydrogen in addition to this ionization which begins at 1 1 volts,
a second and more intense type was found which begins at about 15.7
volts.

4. For mercury vapor no ionization could be detected below 10.27
volts, which is in agreement with the recent work of Goucher and Davis.

5. The ionizing potential of nitrous oxide has been measured and found
to be identical with that of nitrogen. It thus appears that in this case
at least the ionization potential of nitrogen is not affected by its chemical
combination in a compound.

Sloans Laboratory,
Yalb University.

No'^l INTERNAL RELATIONS IN AUDION-TYPE RADIO RECEIVERS, 2 $3

INTERNAL RELATIONS IN AUDION-TYPE RADIO

RECEIVERS.

By Ralph Bown.

THE audion-type radio detector consists of an evacuated glass bulb
containing three electrodes; an electron-emitting hot cathode,
which is commonly a tungsten filament, a cold metal plate placed near
the cathode and held at a considerable positive potential with respect
to it, and, interposed between these two, a grid or lattice of metal wires.
The device is widely used and is well known as a detector in radio-
telegraphy or as an amplifier of electrical impulses such as telephonic
currents. It has been made in various forms and modifications by various
workers, but without radical departure from the fundamental principle
of the control of the thermionic current between two electrodes by means
of the relative electrical potential of a third electrode. A fairly extensive
literature* has been built up about the use of the audion type detector.
Many of its peculiarities and operating features have been fully explained,
but at the same time many of them have not been satisfactorily treated
and not a few of them have been disposed of with the mere statement
that they were due to the irregularities of the conduction of electricity
through gases. The writer has devoted considerable attention to the
effect of the gas in the ordinary audion type bulb and the object of the
present paper is to give some of his results and conclusions. The dis-
cussion is focused particularly upon the interior of the bulb itself and the
relations therein as distinct from the circuits in which the bulb is used,
and upon the explanation of such peculiarities and eccentricities of the
apparatus as may be traced back to the gas.

Theory of Operation.

In the ordinary wireless receiving outfit the circuit used is the one
diagrammed in Fig. i. It consists of three parts which have a common
point at the negative end of the filament. These- three circuits are: the

1 DeForest. Lend. Electr., Vol. 73, p. 385, 1913, or Proc. Inst. Radio Eng., Vol. 2, p. 15,
1914. Reisz, Eleck. Tech. Zeit., Vol. 34, p. 1359. I9i3* or Lond. Electr. Vol. 73, p. 726, X9i3*
Armstrong. Proc. Inst. Radio Eng., Vol. 3, p. 315, 1915, or Lond. Electr., Vol. 74, p. 798, 1916.
Langmuir, Proc. Inst. Radio Eng., Vol. 3. p. 361, 1915, or General Electric Review, Vol. i8,
p. 337, 1915.

254

RALPH SOWN,

filament with its heating battery and regulating rheostat; the plate» in
series with the telephone receivers and the high tension adjustable battery ;
and the grid in series with its blocking condenser, £.C., and the tuned
oscillating circuit coupled to the antenna.

It has been found that the current of electrons from the hot filament
to the plate depends upon the electrostatic potential of the grid in the
manner shown by the curve in Fig. i, and on this as a basis the operation
of the device as a detector of high frequency oscillations has been com-
monly explained in the following manner: Due to the imilateral conduc-
tivity between the hot cathode and a cold electrode, the incoming oscil-
lations are rectified between the grid and the filament and accumulate a
negative charge on the grid and the connected plate of the blocking con-

- o -#•

Ort'd Rai^nfial.

H<lW•-r^l|I|h

Fig. 1.
Ordinaiy audion radio receiver.

F, filament. G. grid. P, plate. B.C., blocking condenser. T, telephone receivers.

operating curve.

a. h.

denser. This decrease of the grid potential causes a corresponding decrease
in the plate current, as indicated by the curve. The dying out of the oscil-
lations allows the charge on the grid to leak off through the gas and the
plate current reassumes its normal value. This function takes place for
every wave train of the damped oscillations and when they occur in rapid
sequence, as from a musical spark transmitter, a musical tone is produced
in the telephone receivers by the changes in the plate current. Oftentimes
the blocking condenser is left out of the circuit and a metallic connection
exists between the grid and filament through the tuning coil. A differ-
ent explanation has been used for such a connection. The grid potential
is supposed to be maintained normally at a point on one of the bends of
the curve such as at (a) or (&). Then as the grid potential alternates
back and forth about this mean value, due to the incoming signals, the
resulting plate current changes, on account of the asymmetry of the curve,
are not symmetrically alternating about the normal value but have a

Na*3. ] J^'^^^^AL RELATIONS IN AUDION-TYPE RADIO RECEIVERS. 255

direct current component. Thus each wave train produces a unidirec-
tional impulse in the telephones and the rapid succession of them gives
the musical tone.

The adjustment of a bulb to procure the best results requires careful
manipulation of the plate voltage and the filament current. Placing it
in a regular receiving circuit, setting for the best operating condition on
actual signals, and then transferring it to a test circuit where the ad-
justments could be duplicated and the data for the characteristic curves
taken, proved to be unsatisfactory, because the adjustment is quite
delicate and can only be correctly made when listening to the signals in
the telephones. Therefore, an artificial circuit was built up as in Fig. 2.
The filament and plate circuits, except for the addition of a voltmeter
and ammeter, were identical with those in Fig. i. In the grid circuit
were placed, an ammeter, a potentiometer with switches to cut it in and
out and a voltmeter to measure the setting, a blocking condenser (B.C.)
with a short-circuiting switch, and a tuned Oscillating circuit, also with a
short-circuiting switch, for receiving signals from the buzzer and auto-
matic telegraph sender in the artificial
antenna circuit to which it was coupled
and tuned. With this arrangement a
detector could be adjusted to the best
operating condition on actual signals
either with the potentiometer cut out
and the blocking condenser in or vice
versa. Then, with the blocking con-
denser short-circuited, the potentiometer
in and with the tuned circuit either re-
ceiving signals or cut out, just as was

Tuned .
Cfrcut

QQ-^-^

"AufiO'Smtteton

lllilMWy

+ re/.
Hi|'l'W«--H'lhQC>J

Fig. 2.
Test circuit.

desired, the curves of grid potential \ ,m,m

against plate current and grid current

could be observed. After taking the

data, or during the process, it was merely

necessary to throw the switches in order

to check back and see that the adjustment as a detector had not

changed. Curves for any other condition than that of best operation

could also be taken with equal ease.

The observations made are here represented partly by the accompany-
ing curves and partly by statements in the text. They show the truth
of the ordinary explanation of the audion working with a blocking con-
denser, throw some new light on the operation without the blocking con-
denser and furnish a basis for a theory of the internal relations in the bulb.

256 RALPB BOWN. [slSSSl

In the curves (Figs. 3, 4, 5, 6, 8), values of current above the zero current
line mean negative electrons flowing to the cold electrode in question
(t. e., grid or plate) while values below the zero current line mean po«tive
ions flowing to the cold electrode. The potential of the common point
at the negative end of the filament is assumed as zero and the grid and
the plate voltages are measured from it. The upper curves show the rela-
tion between grid potential and grid current and the lower curves show
the simultaneous relation between grid potential and plate current. The
voltages labeled on the curves refer to the plate. The real key to under-
standing the action of the audion lies not in the plate current curve but
in the grid current curve and upon it the following explanations are
largely based. The characteristic relations for two points of best ad-
justment as a detector are given in Fig. 3. That Fig. 3 is really a typical

h4.Si<

X>X

a •"'

30.-4V. J

s <^

' ■

^."^

3<S\< ,

^
^

(30.A\f,

OOJ

/ /

Fig. 3.

Fig. 4.

case and that remarks made about it will apply to any similar detector
may be seen by comparison with Fig. 4 which is a composite plot of com-
parable curves taken at random from a large number of audion type
detectors of many different makes and shapes, including some experi-

Na*3^1 INTERNAL RELATIONS IN AUDJON-TYPE RADIO RECEIVERS. 2^J

mental bulbs of exceptional dimensions. Although the curves of Fig. 4
do not lie so close together that they may be said to superimpose, never-
theless, they are all of similar shape and character and in the light of
remarks to follow will be seen to be governed by the same considerations.

It is apparent from Fig. 3 that positive ions exist in the bulb and that
some of them are drawn to the grid, since the grid current crosses and
goes below the zero current line. When a blocking condenser is inserted
in the grid circuit no current can flow through it and so the grid must
assume the potential at which the grid current becomes zero. The squares
on the plate current curves indicate the measured values when the block-
ing condenser was in, and, on comparison with the grid current at the
same ordinates, will be seen to substantiate the above statement. When
a group of voltage oscillations is impressed on the grid, the negative ions
collected by it on the positive half waves far outnumber the positive ions
collected on the negative half waves. The grid acquires a preponderance
of negative charges, assumes a more negative potential, backing off to
the left on its curve and at the same time causing a reduction in the plate
current. When the group of oscillations has passed, the grid is left at a
potential where it is drawing a positive charge, which neutralizes the
former condition. The grid potential moves back to the position of zero
current, thereby allowing the plate current to increase to normal. The
curves show in detail just how this action takes place.

The author has never been able to make an audion work at all well
on the lower bend of the plate current curve and so has eliminated this
from consideration. It was found, however, that good operation without
the blocking condenser could be had not only on the upper bend but in
the straight portion of the curve as well. In fact, in many cases a de-
tector would operate equally well on the straight part of the curve ir-
respective of whether or not a blocking condenser was used and occasion-
ally better without. The explanation of these conditions required some-
thing more than reference to the curvature of the plate current curve.
The required factor proved to be the curvature of the grid current curve.
Take for example the 30.4 volt curve in Fig. 3 or the 39 volt curve in
Fig. 5. With no blocking condenser the grid will normally be at zero
potential. Now if an oscillation be impressed on it, a large grid current
tends to flow oil the positive half waves of grid voltage and
a much smaller current on the negative half waves. The voltage
of the positive half waves is largely used up in resistance and
reactance drop trying to force a large current through the tuning
coil, while the negative half waves, being almost unburdened with
current changes, are free to vary the potential of the grid. The integrated

258

RALPH BOWN.

[Sbcomd
r

effect is to cause the average grid potential during a group of oscillations
to be negative and the plate current to undergo a reduction of the same
sort as occurs when the blocking condenser is used. The best operation
will be realized when the grid potential is normally at a point of most
advantageous curvature in the grid current curve and a battery in the
grid circuit may be of assistance in obtaining this condition. The point
of most advantageous curvature is, roughly, the point where the ratio

..o^

.0/ -

XX>S'

i.O

/.JCam/i.

7-^-^. t^

/ - O -f- /
Grid Pbtenria/''\^o/td.

Fig. 5.

Grid Potent iai' [AslTe.
Fig. 6.

of the current changes produced by equal and opposite potential changes,
is a maximum. This is not necessarily the place where the second
derivative of the curve is a maximum, nor yet is it exactly the same point
for signals of different intensities, because the potential changes produced
by signals are finite and variable. Furthermore, the resistance and
inductance of the tuning coil and the amount of energy the circuit receives
from an incoming signal are factors which affect the relative voltage and
current variations. On account of this complexity in the relations
neither judgment by inspection nor mathematical analysis of the curves
is very helpful in determining just where the best point lies. The tests

Na"i^*] INTERNAL RELATIONS IN AUDION-TYPE RADIO RECEIVERS, 259

indicate that it generally lies quite near the place where the grid current
curve just starts to rise upward from the horizontal. Considering the
curvature of the plate current curve at the upper bend as an explanation
of the operation without a blocking condenser, it is true that a signal
produces a reduction in the plate current of the same nature as, according
to the above explanation, is produced by the action of the grid current
and that the two effects act together for the same result. Two experi-
mental facts lead to the conclusion that the bend in the plate current
curve is a very minor factor. The first is that the operation is not de-
pendent on this bend but may occur just as satisfactorily on the straight
portions of the curve. The second is that the best point is found to be
linked with the grid current bend in the manner explained above. For
example, in Fig. 5 the triangular points are the ones of best operation
without the blocking condenser. They occur just where the grid cur-
rents start to rise and are at the same time well below the knees of the
plate current curves. When receiving loud signals it was very noticeable
that the microammeter in the grid circuit received an impulse with each
dot or dash, in the direction which showed a large momentary excess of
negative ions flowing to the grid.

Knowing that in the average case the detector is working on the straight
portion of the plate current curve and that, therefore, this will only be
of importance as it may undergo slight changes of slope, we may now look
at the shape of the grid current curve as affected by plate voltage and
filament current. We will consider for the present only the facts, leaving
the reasons for later discussion. From Fig. 5 it will be seen that an in-
crease in the plate voltage at constant filament current has the general
effect of shifting the grid current curve downward and to the right.
The 20-volt curves (from another bulb) in Fig. 6 show that an increase
of filament current at constant plate voltage has an opposite effect, the
curve is raised and moved to the left. In order to locate the bend in the
grid current curve at the proper position these two variables must be
correspondingly adjusted. Since they work in opposite directions, an
increase in one is partly compensated by an increase in the other and
vice versa. Thus it is often observed that by raising both plate voltage
and filament current several satisfactory adjustments can be found,
although one of them will usually be most sensitive. The reasons for
the behavior of the curves may be derived from the ionization phenomena

occurring in the bulb.

Action of the Gas.

Langmuir^ has shown that in the absence of any gas the current of
electrons which flows from an electron-emitting cathode to a second elec-

iLangmuir, Phys. Rbv., Sec. Sen, Vol. II.. p. 450. 1913*

26o

RALPH BOWN.

[Sboomd
r

^ 2.0

aav.

Grief fhe^.

Fig. 7.

trode as anode is a function of the geometry of the sj^tem and the voltage
between the electrodes and rs independent of the rate of emission as long
as the saturation current is not reached. He has given curves very similar
in appearance to those of Fig. 7 to illustrate the phenomenon, which is

explained as being due to the " space
charge." The explanation is briefly
as follows: The emission from a fila-
ment is governed by its composition,
its superficial area and its tempera-
ture, according to Richardson's well-
known law.^ As the filament temper-
ature is raised from the point where
emission begins, at first all the elec-
trons are drawn to the anode and the
current is saturated. But finally a
certain equilibrium density of the
negative charges in the space between
the electrodes is reached which masks
the positive charge on the anode to
such an extent that the electric field
gradient at the surface of the cathode is reduced to zero or nearly so.
The negative space charge thus prevents further increase of the anode
current no matter how much the emission may be increased. Now the
audion is dissimilar to the ideal two-electrode system of Langmuir in two
respects; the electric field is greatly modified by the interposition of the
grid member and there is the presence of ionizable gas.

Since the grid is always near zero potential, the electric field between
it and the filament is of very low intensity and, on account of the drop of
potential along the filament produced by the heating current, the field
gradient out to the grid becomes increasingly negative toward the positive
end of the filament. Assuming the absence of any gas, the plate would
thus be restrained from attracting many electrons by the screening
action of the grid, for even though the plate is able to extend its influence
well down between the grid bars, the influence is so weak as to be neutral-
ized by a very small space charge of electrons unless the grid is very coarse
or very high plate voltages are used. It will now be shown how the
introduction of gas and the consequent formation of positive ions tends,
in a certain sense, to nullify the effect of the grid. Some of the ions may
be formed near the filament by collision of positive ions with gas molecules
or perhaps spontaneously due to the high temperature of the gas near the

1 Richardaon, The Emission of Electricity from Hot Bodies, Chapters I. and III.

HSxi^] INTERNAL RELATIONS IN AUDJON-TYPE RADIO RECEIVERS. 26 1

hot surface, but the great majority of them are produced by collision of
electrons with gas molecules in the region between the grid and the plate,
for there the electrons attain the greatest speeds. Irrespective of where
the positive ions come into being, they are drawn toward the grid as the
place of lowest potential. Those outside the grid on being drawn toward
it either strike it and are absorbed or shoot between the bars and bombard
the filament, or, being deflected by collisions, join the number inside the
grid which are more slowly drifting to it under the influence of the weaker
field. There is then, diffusing about inside the grid, an intimate mixture
of positive and negative ions, the presence of both kinds of ions greatly
reducing any effect which may be due to the space charge of one of them.
For this reason the number of electrons which the electric field of the
plate can attract through the grid is greatly increased over the number
which can be attracted when only negative ions are present and their
space charge is fully effective. The grid and the positive ions have, then,
for a steady state, partly offset each other, as is shown by the magnitude
of the plate current and by the previously mentioned similarity of the
curves in Fig. 7, for an audion, to the comparable curves when the grid
and gas are lacking. However, since in the ordinary range the amount
of ionization is practically invariant with regard to changes in the grid
potential, variations in the potential of this member will still have their
full effect in modifying the electric field and consequently the plate
current. The characteristic curves of plate current and grid current
against grid potential can now be accounted for.

PhUe Current Curve. — ^The plate current varies, within limits, directly
with the grid potential because the grid potential (other things remaining
constant) determines the electric field inside the grid and, therefore, the
number of electrons which are drawn out between the grid bars to the
plate.

Grid Current Curve. — ^Since the grid is normally negative all along with
respect to its adjacent filament, the electric field is opposed to its absorb-
ing electrons and it takes on very few. The positive ions are, on the other
hand, attracted to it all along its length, and, as the curves show, it gets
the saturation current of them. Changing the grid potential from
negative to positive causes part of the grid to begin attracting electrons
and the large supply of them allows it to attract a great many. Thus
the grid current curve has a small, nearly constant value below the zero
line for negative grid potentials and rises sharply in the neighborhood of
zero potential as the attraction of electrons begins. The location of the
bend and the absolute values of the grid current ordinates are determined
by the filament current and the plate potential, and also, as will be shown

262 RALPH BOWN. [

later, by the shape of the electrodes and by the nature and pressure of
the gas. Their influence is exercised through their effect on the body of
ions between the electrodes. An equality always exists between the rates
of supply and the rates of removal of both kinds of ions, but the equili-
brium numbers of the ions present and the actual values of the rates will
be dependent on the existing physical conditions as controlled by the
factors mentioned. For low plate voltages very few positive ions are
formed and the charges collected by the grid and forming the grid current
may always consist of a preponderance of electrons (see Fig. 6). As the
plate voltage is raised more positive ions are formed and contribute to
the grid current, while the increased electric field intensity causes the
electrons in the neighborhood of the grid to have a greater tendency to
be drawn between the bars out to the plate and a lesser tendency to strike
the grid, so that a larger positive potential on the grid is necessary to
attract many of them. These two things taken together result in a shift
of the grid current curve downward and to the right. Increase in the
filament current produces an opposite effect for the reason that it raises
the available supply of electrons, thereby increasing the tendency for
them to strike the grid. At the same time it somewhat lowers the
number of positive ions, because the increased number of electrons which
come out from the filament and execute a limited, low velocity flight
inside the grid and back to the filament, is a favorable condition for an
increased rate of recombination.

Looking again at Fig. 7, it will be seen that the best operating point on
the curves is, in every case, just above the knee, as indicated by the
squares. This fact, taken in conjunction with the foregoing discussion,
shows that the most sensitive point, the point where the most advantage-
ous bend in the grid current curve occurs at the zero current value and
where the optimum relation between the flow of the positive and negative
ions to the grid is obtained, is identical with the point where the plate
begins to be unable to draw any more electrons from behind the grid
even if considerably more of them are supplied. For higher filament
temperatures an excess of electrons is present and, although most of them
return to the filament, still, some are forced on the grid even when it is
at a negative potential, which means that the bend is smoothed out and
also, perhaps raised above the zero grid current line. For lower temper-
atures insufficient electrons are present to supply the demand of the plate
and the electric field near the grid is modified so that the grid cannot easily
acquire electrons even when slightly positive. The positive ions form
the principal part of the current and the bend occurs less sharply and
perhaps below the zero current line. Thus, either above or below the

Na3^*] ^^"^^^^AL RELATIONS IN AUDION-TYPE RADIO RECEIVERS. 263

optimum temperature of the filament, conditions are less favorable to
sensitiveness, particularly when the blocking condenser is employed.
This shows why, in the ordinary use of the audion, the adjustment of the
filament current is the final and most delicate one.

The values of plate voltage and filament current necessary for best
adjustment are dependent on the nature and pressure of the gas and on
the dimensions of the electrodes, since these things affect the amount of
ionization and the shape of the electric field. Decreasing gas pressure in
a bulb requires an increasing plate voltage to bring it up to. the best con-
dition. This is often noticed in a bulb which is used continuously for
nome time. The ** clean up " of the gas lowers the pressure and the plate
voltage must be raised from time to time until, finally, either the bulb
must be discarded or the gas pressure restored by heating up the glass
walls. The reason is that the decreased production of positive ions, due
to a reduced number of gas molecules, must be compensated by the
increase of ionization and the shift of the grid current curve which can
be caused by a higher plate voltage. All of the writer's experiments have
been carried on with the residual gas from the ordinary exhausting ap-
paratus, in which case the optimum pressure was .005 to .010 mm. of
mercury. This gas is no doubt made up principally of nitrogen and
water vapor with a trace of mercury vapor, oil vapor, etc. Undoubtedly
changes in the nature of the gas in a tube would have some effect on the
characteristics of the operating curves since they would be accompanied
by changes in the ionizing potentials. Although various gases and vapors,
particularly mercury vapor,^ have been tried in audion-type relays by
different experimenters and with varying success, no data are available
from consistent tests in which similar conditions of electrodes were main-
tained for the different gases. In a bulb which contains ionizable gas
and which is used as a detector, considerable changes in the shape and
size of the electrodes may be made without appreciable effect on the
maximum sensitiveness, because the changes are largely neutralized by
the necessary accompanying alterations in the plate voltage and the
filament current. This is not true of amplifiers containing very little gas.
Variations in the sensitiveness are often observed when a magnetic field
is caused to act on the bulb. These variations are due to the effect of the
field in shifting the paths of the electrons and thereby modifying the
operating curves into more or less advantageous shapes, as the case may
be. Bulbs in which the grid and plate but partly enclose the filament are
most affected by a magnetic field.

An abnormal condition is encountered when the plate voltage is raised

^ Reisz, loc. cit.

RALPH BOWS.

insiderably above the ordinary value. A luminous discharge appears
1 the tube and is seen as a cloud, light blue in color, between the grid
id plate and sometimes extending around the grid toward the negative
id of the filament. It is caused by the active and thorough ionization
[ the gas by electron bombardment. The appearance of the blue glow
often presaged by a hissing in the telephone receivers similar to the
issing of an ordinary electric arc which is running at too high a current
ensity. In bulbs where the filament is only partly screened by the grid
id plate electrodes the glow may, at high volt^es, fill the entire tube,
/ith such raising of the plate voltage the characteristic curves of an
idion undergo radical changes as shown typically in Fig. 8. The

i.5-volt curves are the normal ones on which good detector action is
:alized either with or without the blocking condenser. On the 25-volt
jrves the bulb can be made to work fairly well without the blocking
indenser. Between the 25-volt and 33.S-volt curves it is a very poor
etector though fair as an amplifier, but above 33.S volts it is practically
seless as either. The successive curves occupy positions farther and
Lrther to the left because the screening action of the grid is reduced by
le increasing plate voltage. Not only does the number of positive ions
rawn to the grid become lai^er as the plate voltage goes up but the
lape of the grid-current curve, if inverted, shows a peculiar similarity
) that of the plate-current curve. The two curves are partly interde-

Na'3^'1 ^^^^^^^^ RELATIONS IN AUDION-TYPE RADIO RECEIVERS. 265

pendent at this stage. An increase in the ionization modifies conditions
around the grid, as has been previously explained, in such a way as to
allow a larger plate current, which, in turn, causes more ionization and
consequently still more plate current, so that the conditions tend toward
instability on account of the " progressive ionization." The increasing
eff^t of this phenomenon can be followed in Fig. 8 from the place where
it is present but very slightly (25-volt curves) to the place where insta-
bility is reached and the changes are critical (37.5-volt curves). Blue
glow makes its appearance in the tube at the same voltage at which the
current becomes critical. In discussions of the audion it has occasionally
been stated that the great sensitiveness of the device is due to this pro-
gressive ionization. The author has found no evidence of such a func-
tion in the ordinary range of plate voltages in which successful opera-
tion as a detector may be realized. Even good working as amplifier in
the progressive ionization region is doubtful for in spite of the great
steepness of the plate current curve, the grid current curve is also so
steep that the power amplification is usually poor.

Summary.

Experimental curves are shown from which the details of the operation
of the audion as a detector in radio telegraphy are followed. A theory
of the action of the gas in the bulb is presented which explains the curves
and is in agreement with all the observations. Some of the peculiar
features of operation as influenced by the nature and pressure of the gas,
magnetic fields, the circuits employed, etc., are discussed in their relation
to the theory and the experimental data.

The writer desires to express his thanks to Professors E. Merritt and
F. Bedell for their interest and advice.

Physical Laboratory of Cornell Untvbrsity,
April, 191 7.

266

HARRY Ti BOOTH.

DISTRIBUTION OF POTENTIAL IN A CORONA TUBE.

By Harry T. Booth.

I. Introduction.

I. General Characteristics of D.-C, Corona. — ^The name corona has been
applied collectively to the conduction phenomena appearing when a
sufficiently high potential difference is applied to two electrodes (two
parallel wires, or two coaxial cylinders) separated by a gas. Corona
appears for both alternating and direct impressed potential differences ;

for the purpose of our investigation, however, di-
rect current corona was the more suitable.

Since a knowledge of the distribution of poten-
tial between the electrodes will be necessary for
any fundamental corona theory, an investigation
has been carried out at this laboratory to deter-
mine the field at every point between a wire and a
coaxial tube, under various conditions of impressed
voltage, pressure, size of wire, and current. It is
hoped that the data taken will aid in the formu-

Fig. a.

lation of an adequate corona theory.

II. Method.

The distribution of potential between a wire and a coaxial cylinder
was investigated in the following manner.

A hole was drilled in the side of a cylinder, and an insulated wire ter-
minating in a bare spherical tip was arranged so that it could be moved
radially between the wire and the tube. A micrometer microscope di-
rected on a fixed point of the movable wire served to determine the
relative position of the point. An electrostatic voltmeter of small
capacity was connected in series with the exploring point and the tube.

When the point was moved to any portion of the radial field, the volt-
meter quickly showed a constant deflection, indicating that the potential
of the point was in equilibrium with that of the field at that particular
place.

By moving the exploring point from the tube to the wire, observing
the voltmeter readings at certain intervals, a comparatively accurate
estimate of the intensity of the field was obtained.

Na"3^i DISTRIBUTION OP POTENTIAL IN A CORONA TUBE. 267

III. Apparatus.

1. The Corona Tube. — ^The corona tube as indicated in the accompany-
ing sketch was 35.5 cm. long and 7 cm. in diameter. The central wire
was of copper, well polished, and stretched tightly. In all, four wires
were used, No. 40, No. 32, No. 28 and No. 20 B. & S. gauge.

The ends of the tube were covered with heavy plate glass, drilled for
the central wire, and sealed fast with half and half wax.

Since it was necessary to work at pressures lower than atmospheric, a
glass tube was sealed over the exploring rod, so arranged with ground
joints and springs as to allow the point to be moved at will without
destroying the constant pressure.

2. Source of Potential, — ^The source of continuous potentials used in
this set of investigations consisted of a battery of 40 500 volt, 0.5 ampere,
shunt-wound, D.-C. generators connected in series.

These were arranged so that the potential could be varied continuously
from about 300 volts up to 20,000 volts. Power for the driving motors
was supplied by a motor generator set equipped with a voltage regulator,
so that the voltage variation on the iio-volt power line was constant to
within less than .5 per cent.

In general, the potential of the high tension line was as constant as
the accuracy of the work demanded.

3. Voltmeters, — For the measurement of voltages, three voltmeters
were used, a Kelvin electrostatic voltmeter with three ranges, a Braun
electrostatic voltmeter, and a General Electric electrometer type volt-
meter.

These instruments were calibrated with an attracted disc electrometer,
equipped with a scale and vernier so that the distance between plates
could be read to 0.05 mm. The force on the disc was measured by a fine
balance.

The Braun voltmeter had a range of 0-3,500 volts, and since it is
essentially an electroscope, it was almost ideal for use with an exploring
point.

The Kelvin instrument had 3 ranges, 0-5000, 2,000-10,000, and 4,000-
20,000 volts.

4. Current Measurements. — Currents between the wire and the tube
were measured by means of a D'Arsonval galvanometer, used in connec-
tion with an Ayrton universal shunt. The figure of merit of the galvanom-
eter was obtained, using standard resistances and a dry cell whose
E.M.F. had been determined by comparison with a standard cell.

268

HARRY T. BOOTH,

Table of Curves.

Figure.

Curve.

Wire

Bft8

Gauge.

Voltage.

•

/ Amperea.

PMm.
of Hg.

Temp.
OC.

Remarka.

12.500

9.76. lO-»

745

25**

Faint glow

13,850

6.62. lO-»

745

25**

Good glow

15.420

1.6 .10-^

745

25**

Good glow

16,000

1.78. 10-*

745

25**

Good glow

1.450

3.9 A(n

23.5*

27**

Dull glow

2,150

2.31. 10-*

23.5

27**

Bright glow

2.950

5.58. lO-<

23.5

270

Brilliant purple glow

2,150

Electrostatic curve

10,000

9.23. lO-»

450

27**

3 or 4 steady beads
wire negative

8.400

3.19.10^

745

25**

No apparent glow

10,200

2.66. lO-»

745

25**

Faint glow

11.500

7.1 .lO-»

745

25**

Dull glow

13.450

1.95. 10-«

745

25**

Good glow

14.000

3.73. lO-<

745

25**

Bright glow

1,520

4.43. lO-»

24**

Good glow

1.750

1.35. 10-*

24**

Good glow

2.320

3.73. lO-«

24**

Bright glow

2.890

6.92.10^

240

Brilliant glow

2.320

Electrostatic curve

•

1,800

9MA(n

24**

About 30 steady beads

6.510

4.l7.lO-»

25**

No glow

6.825

1.91. lO-»

26**

Distinct glow

7.425

1.91 lO-»

26**

Good glow

8.400

5.94. lO-»

26**

Good glow

9,900

9.54. lO-»

26**

Bright glow

6,825

1.91.10-*

747

26**

Distinct glow

6.825

2.03. 10-*

241

24**

Bright glow

6,825

3.46. 10-*

885

24**

Brilliant glow

6,825

Electrostatic curve

1 1

5,050

1.79. lO-«

744

26**

No glow

5.650

2.39. lO-»

744

26**

A few dull beads

7,250

3.10.10-*

744

26**

Beads 1 cm. apart

4,520

4.77. 10-*

740

22**

No glow

4.700

1.19.10-^

740

22**

Distinct glow

6.500

2.26. lO-»

740

22**

Good glow

8,400

8.29. lO-»

740

22**

Good glow

9,900

1.67. lO-<

740

22**

Brilliant glow

8,400

Electrostatic curve

Voi-X.
Naa.

]

DISTRIBUTION OF POTENTIAL IN A CORONA TUBE.

269

IV. Results.

I. General Type of Curves. — By the method of exploration already
described, curves for the distribution of potential between wire and tube
were taken for No. 40, No. 32, No. 28 and No. 20 copper wires stretched
along the axes of the tube. These curves were taken for various pressures
and voltages after the appearance of the corona. Representative curves
obtained are shown in Figs, i to 10, and the conditions under which each
curve was taken are given in Table I.

For the No. 40 wire, it was found impossible to obtain curves of the

■ V

11 w^

' \

•\Vv

^ * ^

t \

■

ft V

iV 4.

Fig. 1.

Fig. 2.

potential distribution when the wire was negative; for a given position
of the exploring point the readings of the voltmeter were not constant.
The beads appearing when the wire is negative were seldom at rest, and
this would lead to the conclusion that each movement of the beads is
accompanied by a change in the field surrounding the wire.

For No. 32 wire, when the wire was negative, two curves shown in Fig. 9

■ n*.,

bb»

( '

r^—

'W^

\\\

r \\ \

♦

h \

\'^

."^^

^11

l«R

Fig. 3.

Fig. 4.

were taken before the corona appeared, also a portion of a curve for a
voltage at which there was a distinct series of beads along the wire.

Curves were also obtained for No. 28 and No. 20 wire when the wires
were negative, the same general characteristics being exhibited in each.

3. Discussion of Curves, — ^The corona discharge in general is divided

270

HARRY T. BOOTH.

into two classes, according as (i) the wire is positive, and (2) the wire is
negative.
The first case, when the wire is positive, is characterized by a uniform

' \

«««-

•>

. 1 .

Fig. 5.

Fig. 6.

purplish glow around the wire. The second case, however, differs in
appearance. When the potential is sufficiently high, small tufted beads

• f

ntm-^mftm

A\\

l\\

\V ^

■«•

* X

^iOr-

+-- 1

•*• I ^

Fig. 7.

Fig. 8.

appear on the negative wir^, and are at rest only under exceptional
conditions.

Curves are shown for both positive and negative wires. Let us con*

^ \

TiAay-*.

fc ^

. 1 1

M..

■ 1

nt.'^ttmm

i\\

\ • " \

' r

P^^\

>~^

=4^

M^l

Fig. 9.

Fig. 10.

sider the appearance of the potential distribution curves when the wire
is positive.

nS"3^1 distribution op potential in a corona tube. 271

I. The Positive Wire.

In general, the space between the anode and the cathode may be broken
up into four regions.

1. A region immediately surrounding the wire, which is characterized
by a very large potential gradient. This may be due to the excess of
the number of ions or electrons approaching the electrode over the
number of those leaving, since the former number includes ions generated
at all parts of the field, whereas the latter contain only ions that are
generated in the narrow layer close to the wire. Thus we can see that
the charges on the excess of negative ions near the wire disturb the electric
field so that the potential difference per centimeter, or the gradient, is
large near the surface of the wire.

2. A region of approximately constant force extending from the
" surface layer " region adjacent to the wire, to a point which varies
with the pressure, current, and voltage. At the higher voltages, the
actual potential at a given point in this region is greater than the the-
oretical electrostatic potential, and the tangent to the curve may be
either greater or less. Figs. 2 and 5 show the electrostatic curve (dotted),
in comparison with actual curves taken.

3. A region of little or no force near the tube. In passing from II. to
III. the number of positive ions increases (since they are generated in all
the space between the wire and region III.), and their charges oppose
those on the negative ions to such a degree that not only the negative
charges on the ions, but also the electrostatic forces due to the con-
figuration of the system are neutralized.

4. A region close to the tube, corresponding to the " surface layer "
contiguous to the wire. In this space, positive charges accumulated at
all the remaining parts of the radial field are predominant, and three is
an abrupt cathode drop at the surface of the tube.

2. Wire Negative.

When the wire is negative and corona appears, a potential curve is
obtained which differs somewhat from the positive curves. Large
cathode and anode drops appear, and the intervening space has a very
small field. Reasoning similar to that explaining the shape of the curves
when the wire is positive explains the negative curves.

So in general, the anode and cathode drops of potential are predominant
in both types of curves. There are several reasons for this, namely:

1. Polarization potential between a metal and a gas.

2. Accumulation of ions.

3. Reflection of ions.

272 HARRY T. BOOTH. {:

4. Different velocities of positive and negative ions.

5. A non-uniform field.

The Potential Curves from a Theoretical Point of View.

I. The starting point of the corona.

We have Peeks empirical formula for the starting intensity,

^-^•(^+^)

(I)

where £1 is tiie force at the surface of the wire of radius Ri and Eq and
fi are constants.

From the general electrostatic theory, at the moment when the corona
discharge is starting, just before the field has been disturbed by the moving
charges,

Therefore at the instant when the corona starts

which resembles the general formula for the electric force between two
concentric cylinders.

Hence, when r = i?i + fiy/Ri,

E = Eq,

2. Calculation of the volume density of electrification in the space
between the two concentric cylinders.

For a system where the potential at a point is due to moving charges
as well as static charges, we have Poisson's equation expressing the
density in terms of the potential,

V^7 = - 4^P, (6)

or, writing it in cylindrical co5rdinates,

dW I dV ^ I dW dW

Na3. J

DISTRIBUTION OP POTENTIAL IN A CORONA TUBE.

273

For this particular case, the derivatives in z and i- are zero, so rewriting
the above equation, using total derivatives.

^ idV
dt^'^ r dr

— 4Tp,

(8)

Since the density is an undetermined function of the radius, the equa-
tion cannot be integrated directly. If, however, we plot the potential
against the distance from the axis, a graphical method will aid in the
determination of the density. That is, if the first derivative of the
potential is determined from the curve for a series of values of r, these
new values may be plotted against the radius again. By repeating this
process with the derived curve, a relation between the second space
derivative and the radius is obtained. From these two derived curves,
then, the density may be computed according to equation (8).

Fig. II is a repetition of Curve 4, Fig. i, and Fig. 12 shows the density
as computed for the different values of r .

The density curve shows what we have deduced intuitively in regard

' \

1 "^

•• ?

1 —

Fig. 11.

Fig. 12.

to the charges necessary to produce the observed distortion of the field.
The large resultant negative charge near the positive wire and the positive
charge near the negative tube should be expected. A peculiar maximum
ai^)ears at about 2.7 cm. from the wire (Fig. 12).

4. Sources of Error,

I. Potential assumed by a sphere in an ionized gas.

It is difficult to draw conclusions as to the absolute potential of a sphere
in a .conducting gas, since it is very likely that the potential at an undis-
turbed point in a gas is not the same as the potential assumed by ^ sphere
when its center is at this point.

In the case of a sphere near the positive electrode, its potential being
initially the same as that of the gas, two streams of ions move in opposite
directions past the side of the sphere, one containing a large number of

274 HARRY T, BOOTH. [ISb5!

negative ions, and the other a smaller number of positive ions. It
intercepts more negative ions than positive, so that its potential falls
below that of the surrounding gas. The charge thus acquired by the
sphere increases until the effect which it produces in attracting positive
and repelling negative ions causes them to come in contact with the
sphere in equal numbers. The final value of the potential assumed by
the sphere is too high by an amount which depends upon the relative
velocities of the positive and negative ions.

Conversely, when the exploring sphere is close to the negative electrode,
there are a greater number of positive ions intercepted than negative ions,
so that the potential of the sphere rises above the potential of the undis-
turbed gas, until finally an equilibrium is reached, the number of positive
charges acquired by the sphere being equal to the number of negative
charges. Thus the potential assumed by the sphere is greater than that
of the undisturbed gas.

If, however, the velocity of the positive ions is approximately equal
to that of the negative ions, then the exploring point should attain very
nearly the same potential as that of the surrounding gas. For the
pressures used in this series of experiments, the velocities of the ions are
nearly the same. Thus the error introduced could not have been very
great.

A slight error might be introduced if there was an appreciable voltmeter
leakage between the point and the power line. The shape of the point
also affects the shape of the potential curve to a small degree. The volt-
meters used were practically free from leakage, and the work was done
during cold, dry weather, so the error introduced from this cause is neg-
ligible.

An attempt is being made to formulate the mathematical theory of the
corona discharge, and it is hoped that these potential curves will aid in the
solution of the problem.

Summary. — ^The distribution of potential between the electrodes of a
corona tube was determined for four sizes of wire, for various pressures
and potential differences. From these curves the density of the charge
along the radius was derived by means of graphical methods.

In conclusion, I wish to express my appreciation of the suggestions and

advice given by Dr. Jakob Kunz, of this laboratory, and to Mr. J. W.

Davis and Mr. R. W. Owens for the use of portions of their data on this

problem.

Physics Laboratory.

University of Illinois,
May II, 1917.

Jgj-^] EFFECT OF STRAIN ON HETEROGENEOUS EQUIUBRIUM. 275

THE EFFECT OF STRAIN ON HETEROGENEOUS

EQUILIBRIUM.

By E. D. Williamson.

FOR some time past Mr. Hostetter of this laboratory has been carrying
on experiments (soon to be published) on the solubility of stressed
solids, the results of which are so little in accord with the theoretical
views which are commonly held that a careful scrutiny of the assumptions
which underlie the various theoretical discussions and of their con-
sequences seems to be needful. This particular case has not, so far as I
know, ever been thoroughly discussed from the mathematical side except
by Gibbs, whose treatment is difficult to follow. Other writers have
contented themselves with reasoning from analogy and using equations
derived for other special cases. There seems to be no point in discussing
all that has been written on the different parts of the problem as two recent
writers have summarized the bulk of it and a reference to them seems
all that is necessary.

Johnston^ has discussed the effect of '* unequal " pressure on melting
in three different papers.* He makes no reference to his premises but
uses a formula given in slightly different form by Poynting.' According
to this formula the effect of " unequal pressure " is very great; for ice it
has twelve times the effect of hydrostatic pressure on both phases. In
the third paper cited he remarks: '* Considerations in every respect
analogous to the foregoing are applicable to systems of a solid in contact
with water or other solvent; in such cases, pressure acting in excess on
the solid phase increases its solubility" and in a footnote adds: "The
amount of this increase of solubility can be computed from equations
analogous to these applicable to melting points."

In the present paper we hope to show that while his statements are
right qualitatively, his quantitative deductions are incorrect owing to the
inapplicability in his cases of the equation used.^

'J. Johnston, Jour. Am. Chem. Soc., 34* 789, 1912; also J. Johnston and L. H. Adams
Am. Jour. Sci., J5. 205, 1913; and J. Johnston and P. Niggli, Jour. Geol.. 21, 602, I9i3«

* " Unequal " pressure has unfortunately been used in two senses, viz., (i) a difference of the
hydrostatic pressure on two phases, and (2), a one-sided stress on a solid, and equations derived
for (i) have been indiscriminately used for both.

» J. H. Pojmting. Phil. Mag.. $. 12, 32. 1881.

* See previous footnote.

Z7(> E. D. WILLIAMSON. [swaw.

More recently Bridgman^ has deduced a formula of very formidable
appearance dealing with the change of melting point or transition point
with stress. He unfortunately also makes no mention of assumptions,
giving as his reason for this " The formulas were derived by ordinary
thermodynamic methods; it is hardly worth while to reproduce the
wearisome details." As regards the mathematical transformations this
is true, but we hope to show that several of his terms rest on very shaky
foundations. Complete references to earlier literature is found in
these papers and hence no more citations are necessary here.

From the point of view of the phase rule the introduction of a stress
on one of the solid phases simply means that a new variable appears in
the equilibrium equations. We can therefore have one more phase
under given conditions or conversely the system has one more degree of
freedom with a given number of phases. For instance in a one-component
system the coexistence of three phases is no longer sufficient for invari-
ancy; there is a possibility of the presence of four co^stent phases, say,
vapor, liquid and two forms of solid. The method of treatment in this
paper will be on the lines of finding the form of the term containing the
new variable and the consequent change in the equilibriiun equations.

In view of the laxity in the statement of funciamental hypotheses in
the papers already referred to it seems wise to deal with a simple case
first, in order to make perfectly plain whence the differences in existing
formulae arise. As the mere mathematical transformations are simple the
treatment can easily be extended to cover the general case. In this
paper, therefore, we deal first with the simplest possible case, viz., the
effect, on the melting point of a solid, of a unidirectional thrust, and the
extension to more general cases will be briefly referred to. The notation
used is that of Gibbs except when specially defined otherwise.

Suppose then that we have a cylinder of solid in contact with its
melt and that on one end of the cylinder a thrust t dynes per cm* acts
in addition to the hydrostatic pressure p. Let ^** be the temperature.
It is required to find the relation connecting the changes of ^, t and 6
on the assumption that we are dealing with a reversible equilibrium — ^a
discussion of the evidence of reversibility is given after the mathematical
transformations have been dealt with. We also assume that the strain
throughout the cylinder is uniform. As we need only consider an in-
finitesimal amount of the solid at the particular surface under discussion
at the moment, this is not really a limitation.

Let M be the energy required to bring unit mass, of the substance
considered, from a fixed reference state into the system without change

» p. W. Bridgman. Phys. Rbv.. VII., 215. 1916.

b '

No!"3^1 EFFECT OF STRAIN ON HETEROGENEOUS EQUILIBRIUM, 277

of volume of the system, i. e., without the system doing work against the
outside forces, the reference state being our arbitrary zero entropy state.
Then, as Gibbs^ showed, m and the changes in fi must be the same for all
parts of the system so long as a reversible equilibrium state exists.

Therefore in our case equating the changes in m for the same change in
p and 0 for liquid and solid and a change in t

(The left hand subscripts in this expression refer to the phase under con-
sideration, L for liquid, 5 for solid.) It is unnecessary to put mass among
the variables, as /* is evidently independent of the actual mass of the
substance which is present as solid or liquid. (This would not be trqe
for the case of solution, as in this case we have to consider the change of
concentration in the liquid phase.) To get the required relation it is*
therefore necessary to evaluate the partial differential coefficients for
each phase. For the liquid we have for any change

d€ = edfi — pdv + fidtn [e = energy, 1; = entropy]

.*. d{€ — Sri + pv) = — ride + vdp + ydm

•••.Cll-ja^L-"' !>-. -P«- V0..I

and

j\deJp \dmJp^0

For the solid, on the other hand,

d€ = Bdrj — pdv + fidm — vAdh
where A = cross section of, and h = height of, the cylinder.

Now

A =' =^^'

h h
where V, = spec. vol. of solid

/. wAdh = irmVsd{logh)
.*. d€ = Bdfj — pdv + ydm — irmF«d(log h)
:. d(« — ^ + /w + vmVs log h)

- — ride + vdp + fidtn + log h{irmdV, + wF^t + irVtfim).
Let dV. = aV^e - PV^P - yV^r, so that a = (i/F.) (dV./de) «

1 ThoM familiar with Gibbs will notice that we have slightly extended his use of m* but
this does not affect our statements.

278

E. D. WILLIAMSON.

rSaco
Isbu

coeff. of expansion with temperature, and similarly j9 and y are the
coefficients of compression.

Put in this value for dF„ collect like terms and the result is

d(€ — Sri + pv + irmF, log h) = (aTwF, log h — ri)de

+ (» - pTTfnV, log h)dp + (m + tF. log h)dfn

+ (mVt log A — yVtTfn log h)dir

From this we evaluate the necessary partial differential coefficients, e. g.,
— (m + tF, log A)„ p. m « ^ (aTwF. log h - iy)p, ^, «

whence

Similarly

/alogA\
+ airmF, I ^.^ I

,\ dm /p, w, •

d log A

• AOT/p, # c^ v^ /p, •, m t^ vW 'Pf». •

and

Adpf,,$ •\dw/„p. , \\ d^ /„,,i,, ^ '.X dm /„p,.

We have now evaluated all the necessary terms for substitution in our
original equilibrium relation. This gives us

\(^) .{^) _ ^j^ + p^v; {'-^) ]dp

"^ I L\dm},,,'^.\dm}„„,'^

a-wV,

-axtnV. (iMi) 1

' »\ dm /p, „ # J

<f9

-[•"'• + <■ --)«^{^l...]^

Here a/3 has been substituted for

I idh\ idlogh

I idh

h\de

) - (-^)

/ p, V, m ^ OU / p, ,, m>

- j8/3 for i/h(dh/dp) and ^ for - (i/h){dh/dir) = reciprocal of Young's
modulus.
A number of the terms still need explanation. All the differentiations

Noa^'l EFFECT OF STRAIN ON HETEROGENEOUS EQUILIBRIUM. 279

left are with respect to w with p, w and $ constant, i. e., they are the
changes which occur when heat is added so as to melt a little of the
substance under constant conditions.

and

_ (ii) + (£1) ._h

(where X = the latent heat of the solid under the conditions signified),
(d log A/dm)p, ,, t is a term which is different according to the face from
which the solid melts. Since />, t and 6 are constant the state of strain
is constant and therefore the value of the term is zero^ unless the substance
melt from the face which is being thrust and in this case the point of
application of t moves down through the space occupied by the part
which melts. This gives obviously

dh dm
h " m

diXog h) ^ I
dm m

Our equilibrium condition therefore reduces to
(i) at free surface where(d log h/dm)p^ », • = o

(2) at thrust surface where (d log h/dm)p^ », • " iM

= leirV. + {i -yT)V.]dir B.

Next we extend these formulae and will then discuss the assumption of
reversibility.

First consider the values of {d$/dv)p, neglecting the terms containing
the elastic coefficients; a step which is permissible unless for very large
pressures and in that case we get permanent deformations and our
equations no longer apply. Case (I) A reduces approximately to

dd eTrVadT work done on unit mass

d X "" latent heat

Integration gives us log

^ This equation was also got by E. Riecke in 1894. Ann. Phy9.» 54, 731, 1895.

28o

E, D. WILLIAMSON.

rskcoMD

It is evident that we can extend this and can state as a general theorem
that the melting point at the free surface is depressed by any stress by
an amount dependent on the latent heat and on the work done on unit
mass. (Cf. Case 4 in the above-mentioned paper by Bridgman.) Case
(II.) B reduces approximately to

de _ V^T
e " X '

, e v.TT

(approx.).^

Therefore at the thrust surface the melting point is depressed by pressure
but raised by a tension, being now dependent on t and not on ir*.
The amount also is much larger than that for the free surface. (Cf.
Bridgman as above.)

The value of (dd/dp)^ is approximately and, for t = o, exactly that
given by the Clausius-Clapeyron equation.

The values of (dp/dTr)^ differ similarly to the (de/dirjp values. That
is to say — suppose we have ice at — 5® C, we know that a certain hydro-
static pressure will melt it, but if we put a thrust on the ice and then melt
it with hydrostatic pressure we shall need a much less pressure if the ice
can melt on the thrust surface but only a very slightly less pressure if
the melting must take place on the free surface.

Extension to Solutions.

It is a simple matter to extend our formulae so as to cover the case of
the solution of a salt in water or other solvent. The only change is the
addition of an extra term to our equation, as m in the liquid is a function
of the concentration, so that we must have a term (dfi/dm)din on the
liquid side. The mass of solvent is supposed constant, as that involves
no loss of generality. We have now four variables and can therefore
choose two to be kept constant.

Consider the case of p and $ constant. The general equation reduces to

Case I., (A)

Case II .,(B)

(T— I dm = erVtdT
_ am / p, #

l^JL\

dm = V^T

(approx.)

(approx.).

Where dm is the amount of salt which must be dissolved in a fixed amount
of the solvent in order to keep the system in equilibrium under the in-
creased stress on the solid and (dju/dm)p, « is the rate of change of /i in
the liquid for such a change of concentration.

^ This equation ia the one used by Johnston.

)

No'a?^'! E.FFECT OF STRAIN ON HETEROGENEOUS EQUILIBRIUM. 28 1

Now {dfildfn)p, 0 is always positive (see Gibbs) and we therefore have
the theorem that the solubility of a salt is always increased by a small
amount on the free surface, but that on the thrust surface we get a rel-
atively large effect, which is positive or negative for a thrust or a tension
respectively. To get the exact values we evaluate the term x(d/*/dw)p, •
For dilute solutions, according to Gibbs, it is equal to RB/M X i/m where
M = molec. wt.

For concentrated solutions we can find its value if we have the necessary
experimental data to get the slope of the solubility curve, with respect
to temperature or hydrostatic pressure, of the unstressed solid and the
necessary heat or volume changes. For if the solid be unstressed our
general equation reduces to

whence

(a..-.-/(5l-<'''-^'V(E)

The denominators are the slopes of the ordinary solubility curves. Here
Vl is not the specific volume of the whole liquid but only of the component
considered and is therefore got by dividing v (the total volume of the
solution) by m (the weight of salt in solution), or in other words it is the
reciprocal of the concentration while X is not the latent heat but the heat
of solution of a gram of the salt in the saturated solution.

Discussion of Assumptions.

The fundamental assumption made is that of reversibility which is a
necessary premise to the equality of the potential (m). This assumption
needs some explicit criticism and justification.

A physical conception of case A can be got as follows. Suppose that
a cylinder of the solid is clamped between two immovable plates so as to
give the necessary stress (see Diagram I.). If solution or melting takes
place round the free surface the conditions will be exactiy right as the
strain will remain constant. If now the action is reversed the solid will
deposit and the molecular forces are such that the state of the new ma-
terial will presumably be continuous with the old. Experiments in this
laboratory with crystals and solution indicate the truth of this.

The supplementary assumptions for the second case (B) may be dif-
ferentiated as follows. In Diagram (2) let / represent the initial state,
A' the final state in case A and B' in case B, when a portion of the solid
melts at constant values of p, t, $. In A' the cylinder has become thinner
but not shorter for t, p and $ are constant in the process. In B\ however.

£. D. WILLIAMSON. [&SS

irface of application of the thrust hag moved down and work

I — A) or «T (where v - volume of part which melted) his been

The question is " Has this work been performed reversibly? "

Fig. 1. Fig. 2.

t following obvious objection may be made. The probable course
iction for this case is that a layer of the top surface melts and the
>d push then descends, merely pushing the liquid out of the way and

no work on the system. If this be so it means that the work
) — h) plays no such part as we have supposed in the mathematical
rtions but is merely dissipated as heat with a consequent lowering
I actual amount of heat we have to add to the system from the heat
roW in order to melt a given amount of the solid. This requires an
Ltion in our original equation as irAdl was used for the differential
term and dl must now exclude any distance moved owing to flowing

of the melt and include only the amount of motion necessary to
ice the state of strain and that momentarily undergone while the
changes to liquid.

this case the troublesome term (d log A/dwi) ,, , , o may be evaluated
proximately (F^ — V^rnVg and substitution gives us a value for

which is approximately the same as the Clausius-CIapeyron one.
is would apply strictly if the liquid remained under the push v
[1 any actual case it is probably pushed out irreversibly, as we have
sted, and there is no actual equilibrium of the kind supposed in our
1 assumptions. We can only hope for a kinetic approximation to
3rmula and cannot test it statically. No exact treatment of such
ems as that of a weighted wire melting its way through ice or the
ng of snow under the runners of a sledge is possible, although many
been given without any qualification. (Cf. Bridgman's case 5.)
idgman's general formula is exactly similar to but more complicated
that deduced in case B. It could be very much simplified by leaving
he terms dependent on the above-mentioned doubtful hypotheses,
e following qualitative conclusions can be drawn:

Na"3^] ^PP^CT OF STRAIN ON HETEROGENEOUS EQUILIBRIUM. 283

I. A stressed solid on contact with a liquid phase containing it is in
metastable equilibrium as it is unstable with regard to the formation
of unstressed solid. The phenomena of undercooling and of super-
saturation however make the state realizable.

II. Such a stressed solid in this state of metastable equilibrium will
adjust itself so that the stress is uniform at least over the surface exposed
to the liquid and in the melting point case the stress on the surface will
be the maximum stress, as if any part inside was at a greater stress it
would necessarily melt and readjustment would take place. If a solid
near its melting point be stressed, therefore, local melting will ensue
with redistribution of the stress. Exactly how far below its melting point
such phenomena will take place must remain an open question as we do
not know exactly how much stress individual grains of a solid may be
subjected to in operations such as bending.

In conclusion it is necessary to add that one argument in favor of the
possibilities of a reversible action even on the thrust surface has been
suggested to me. Becker and Day^ found that growing crystals can
raise weights in their growth even when perfectly free to grow out side-
ways. In these cases the molecular forces were apparently such that
they force the weight upwards in order to deposit material on the surface
in question. What the course of the action was seems far from clear.
Dr. Day has suggested to me that the properties of a film of adsorbed
liquid upon the thrust surface where the growth takes place operate to
modify the condition assumed.

The experiments of Hostetter mentioned at the beginning will clear
up some points and we hope that further experiments now under way will
produce additional evidence. Until such evidence is forthcoming it is
not wise to dogmatize about doubtful points.

Summary.

Equations have been deduced connecting the changes of the variables
which determine the equilibrium between a stressed solid and a liquid
phase. These equations are grouped in two classes A and B; A referring
to a free surface and £ to a thrust surface. The assumptions for case
A seem justifiable but those for case B seem very doubtful and these are
the assumptions upon which, without adequate discussion, nearly all
previous formulae have been deduced.

Experimental evidence is necessary and is being sought.

Gboprtsical Laboratory,
Carnbgib Institution,
Washington, D. C.

* G. F. Becker and A. L^Day, Proc. Wash. Acad. Sci.. VII., 283, 1905.

284 ARTHUR WHITMORE SMITH. f^S

DEMAGNETIZATION OF IRON.

By A&THxm Whitmorb Smith.

THE magnetic state of a piece of iron depends not only upon the field
intensity to which it may be subjected at the moment, but also
upon the previous magnetic history of the iron. This is especially true
when the applied magnetic field intensity is small. In making a magnetic
test of a bar or ring of iron to determine either the B-H induction curve
or the fjtr-B permeability curve it is necessary to use some means to
destroy the effect of previous magnetization. Usually this is done by
the method of reversals, in which the magnetizing current is reversed
many times while at the same time it is gradually reduced to zero. An
alternating current^ is often used for this purpose, but when used to demag-
netize solid bars it does not produce the desired result, as is shown below.
It is more effective* to use a direct current reversed by hand not faster
than one cycle per second while it is slowly reduced, and even then it
has been foimd that the iron is not brought to a constant condition until
after a few hundred reversals of the same value of the current.

In a former paper* it is shown that the magnetic flux does not reach its
full value until several seconds after the reversal of the magnetizing
current. This is due to eddy currents which circulate in the iron in a
direction opposite to that of the applied current, and only after sufficient
time has elapsed to allow these eddy currents to die away will the effective
magnetic field within the iron reach its full value. It is evident, then,
that if the magnetizing current is reversed too rapidly there will be very
little effect upon a considerable (interior) portion of the iron, and the
demagnetization will be less effective than a few reversals made much
more slowly.

The object of the present investigation is to determine the effect of var-
ious methods of demagnetization on an iron ring. Suppose, for example,
that it is desired to find the magnetic induction, B, corresponding to a
given (small) magnetic field intensity, H. If the iron has been previously
magnetized it will first be necessary to completely demagnetize it. This
treatment should be complete enough to wipe out every trace of perma-

» See Searle. Jour. Inst. Elec. Eng., Vol. 34, p. 61, 1904.

* See Burrows. Bull. Bureau of Stand.. Vol. 4, p. 205, 1908.

* Smith. Phys. Rev., Vol. 9, p. 419, 1917.

J}^3^'] DEMAGNETIZATION OF IRON. 285

nent or residual magnetization, and leave the iron equally ready to receive
the new magnetization in either direction. There are thus two questions
to be answered ; first, in what state will a given process of demagnetization
leave the iron? and second, in what manner will the B-H magnetization
curve, or the fir-B permeability curve, be affected by the method used to
demagnetize the iron?

Four different methods of demagnetization were tried. In the first
alternating current was used. In the second the magnetizing direct
current was reduced to zero without reversal and no further demagnetiza-
tion was attempted. In the third method the iron was carefully and
completely carried through a most thorough process of reversals and one
that has been considered sufficient to entirely demagnetize the iron.
In the fourth process the current was reversed fewer times, but with much
longer intervals between successive reversals.

To show the effect, or rather the lack of effect, of alternating current
the iron ring was subjected to a 60-cycle alternating current. The
initial current was 1.5 amperes, and this was slowly reduced to .02
ampere. The maximum previous magnetization was with 1.5 amperes
of direct current, and the maximum of the alternating current should be
well above this. If the iron followed this decreasing magnetic field it
should have very little magnetism left after this " demagnetization."
Whether it is demagnetized or not, the ring being a closed magnetic
circuit gives no external evidence of its condition.

The hysteresis curve for this ring is shown in Fig. i. The maximum
value of jff, the magnetic field intensity within the iron, was 13.3 gausses,^
that is, 13.3 gilberts per centimeter. This corresponds to a-current of
1.5 amperes in the primary winding of the ring. The last application
of the direct current was + 1.5 amperes and the circuit was then broken,
leaving the iron at £, Fig. i, just before using the alternating current.
Afterwards the same value of the direct current was applied in the reverse
direction. Had there been no intermediate treatment this reverse field
would have carried the iron from E to jD, thus completing half of the
hysteresis cycle. The actual deflection of the galvanometer was 109,
instead of 119, showing that the alternating current did have a little
effect and had reduced the average residual magnetization from OE to
OR, Fig. I. This means that three fourths of the residual magnetization
was not removed by the alternating current.

No larger current was used at this time, but after all of the other
measurements mentioned in this paper had been completed and the ring

^ This word is used in accordance with the recommendation of the International Electrical
Congress. Paris, 1900. See The Electrician (London), Vol. 45, p. 822, 1900. Searle, Jour.
Inst. Elec. Eng., Vol. 34. p. 56, 1904. Wol£f, Bull. Bureau of Stand., Vol. i, p. 49, 1904.

286

ARTHUR WBITMORE SUITE.

had been demagnetized several times by direct current reversals, the
alternating current was tried again. Demagnetization by gradually
reducing the alternating current from 1.5 amperes to zero, after the iron
had been subjected to a direct current of 1.5 amperes, gave practically
the same result as before. When the iron was magnetized in the opposite
direction the alternating current removed the same fraction of the residual
magnetization. This was not the case when the alternating current was
reduced by withdrawing a pointed copper electrode from a jar of water
in which was a larger &xed electrode. This arrangement caused a partial
rectification of the current and gave
it a greater demagnetizing effect in
one direction than in the opposite.
When the current was reduced by
metallic resistances no asymmetry
was detected. .

Somewhat better demagnetiza-
tion was obtained by using larger
currents. The largest current used
was 16 amperes, which would cor-
respond to a maximum field of 200
gausses within the solenoid without
the iron. After magnetizing the
iron to D, Fig. i, with a direct field
of 13.3 gausses this lai^ alternat-
ing current, gradually reduced to
zero, brought the iron Xo Z, Fig. I,
and left it with a residual mag-
netization of 4,000 maxwells per
square centimeter. Computation
shows that about ten times this
current would be required to com-
pletely demagnetize the iron, but the copper wire used for the magne-
tizing solenoid would not carry so lai^e a current and it was not tried.
As is shown later, see Fig. 2, if it were assumed that the iron was com-
pletely demagnetized, and a5-/f curve is determined by the usual method
of reversals the measured values of B will be too small and the resulting
curve is not the normal magnetization curve.' Moreover, there would be
nothing to indicate that the curve does not truly represent the iron of the
ring, or that the iron was not properly demagnetized at the beginning.
When the magnetizing field ts reduced from its maximum value to zero
' Sn, also. BuiTOwa. Bull. Bureau of Scand.. Vol. 4, p. 310, Figs. 4 and 5.

,™3

1 m

■ y

r,^

<!»••

Fig. 1. ,
HyM«r«ii curvM lor a ling of Swedlib
Iron. Dtmognetixation by alternftting cur-
rent leaves the iron at R. At one cycle per
iecond the iron la left at L.

^^] DEMAGNETIZATION OF IRON. 287

without reversal the iron is brought to E, Fig. i. This would not be
considered as demagnetization at all, but this point is only a little higher
than R, where the iron was left by the alternating current.

In the third method the iron ring was demagnetized by using direct
current reversed by hand with a mercury commutator at the rate of one
cycle per second. At the same time the current was slowly reduced from
1 .5 amperes to .05 ampere. Reversing this small current gave a deflection
of 2 scale divisions for the total height of the small hysteresis curve.
Applying the full reverse current of — 1.5 amperes, thus carrying the
iron from the lower tip of the small hysteresis curve, L, Fig. i , to the lower
tip, I>, of the large hysteresis curve, gave a deflection of 103 divisions.
This means that the center of the small curve L is at 5 — 5,000, and
thus this demagnetization has left two thirds of the residual magnetization
undisturbed.

In the above case the current remained at +1.5 amperes for a few
minutes before the reversals began. The same process was repeated,
but the current was allowed to stand reversed at — 1.5 amperes for a
short time before beginning reversals. The current was slowly reduced,
as before, and the small hysteresis loop for the last reversal of .05 ampere
is shown at Af . Thus the iron may be left at L or Af , depending upon the
trivial (?) incident of which way the current was flowing when reversals
began.

Of course this is readily explained in the light of the former paper^
on the time lag of magnetization. Owing to eddy currents in the body
of the iron the interior parts are shielded from the effects of the applied
current changing at the rate of one cycle per second.

It therefore seems logical to try the demagnetizing effect of a current
reversed as slowly as once in ten seconds. The magnetizing current was
reduced by twelve steps of such magnitudes as to reduce the induction
by approximately equal amounts. At each step the current was reversed
once in ten seconds for ten times. Finally the full current was applied,
and the iron was found to have been very near to the point 0, Fig. i.
Upon repeating this several times there seemed to be a slight effect in
favor of the direction of the initial current. The number of reversals
at each step was then increased to eleven, which meant that the current
was reduced first while H was negative, next when H was positive, then
when H was negative, and so on. This left the last small hysteresis loop
symmetrical about 0. With only three reversals at each step the de-
magnetization was very satisfactory, and with one reversal at each step
the demagnetization was more nearly complete than in any case where
the current was reversed as often as once a second.

» Smith, Phys. Rev., Vol. 9. P- 4i9. 191 ?•

288 ARTHUR WHITMORE SMITH, \^Smi

The mere fact that the iron is at 0 is not sufficient proof that it is in
the neutral condition. But if it is not at 0 it is certain that the iron is
not demagnetized. And if the iron has been brought to 0 by many
reversals of the current it should also be in the neutral state.

The particular times mentioned here apply to the particular ring em-
ployed, which had a cross section of 4.9 sq. cm. of iron. When using
iron of greater section the time effect is proportionately increased, and
the current should be reversed more slowly. With iron of smaller section
the same demagnetizing effect can be obtained when the current is re-
versed more rapidly.

In view of the fact that the process of demagnetization by reversing
the current as often as once a second does not actually leave the iron in a
demagnetized condition, the question arises, what sort of a B-H curve
would be obtained by the usual method of reversals? And how would this
curve differ from the B-H curve obtained after the ring has been com-
pletely demagnetized?

In the first place it may be noted that the reversal of the smallest
current carries the iron around the small hysteresis loop L, Fig. i, instead
of th^ corresponding hysteresis loop at the origin. While the abscissae
of these two loops are the same, the loop at L is less inclined to the
if-axis, and therefore the galvanometer deflections will not be the same
in the two cases.

Had the magnetic field been simply reduced to zero from its maximum
previous value, and then carried through the same cycle of values, the
hysteresis loop for the iron would have been at T, and the galvanometer
deflection would be still different. From the deflection alone it is not
possible to determine which of these three states is being measured.
Some information, however, may be obtained by applying the full value
of Hf thus carrying the iron from the state in question to D. The cor-
responding change in B is measured by the galvanometer.

If the iron, after being carried around the loop T, is subjected to a
slightly larger value of jff, and again carried around the hysteresis cycle,
the new loop S will be somewhat below T, but it will be as high as possible
and still lie within the larger loop FD, and no amount of reversing the
current will bring it to 0. Larger loops will extend below 0 as shown by
PQ. These loops were experimentally determined for this ring. The
final loop will be FD, and this is the only one which is symmetrical about

fl 0.

' When the iron has been completely demagnetized and then carried

through a similar set of magnetizing cycles, starting with the iron at 0

instead of at T, all of the loops are symmetrical with respect to 0, and

the last loop is FD.

VOL.X.1

DEMAGNETIZATION OP IRON.

289

To find the effect that the method of demagnetization has upon the
magnetization curves, these curves were obtained following three different
methods of demagnetization. In the first the current was reversed at
one cycle per second while at the same time it was slowly and gradually
reduced from H '^ 60 gausses, which was the highest field the iron had
experienced up to this time. The smallest current was then reversed
and the corresponding induction change measured by the fluxmeter-
galvanometer which has been previously described.^ Then the next
largest current was used, and so on
until the maximum current was
reached. The results, expressed in
terms of the tir-B permeability curve,
are shown by Curve I., Fig. 2.

Again, the iron was magnetized to
fl'=6o, and the circuit broken. This
leaves the iron at a point correspond-
ing to £, Fig. I. For the sake of
comparison the same set of measure-
ments was repeated without any
further demagnetization of the iron.
The results, expressed in the same
way as before, are shown by Curve
II., Fig. 2, which, as would be ex-
pected, lies below curve I.

Finally, the iron was subjected to
a complete demagnetization by re-
versing the current once in ten seconds while the current was reduced a
little after each third reversal. Of course this process was long and slow.
The induction measurements were made in the same manner as for the
two previous cases, and the results are shown by Curve III., Fig. 2. This
indicates a higher permeability than Curve I. In fact, there is more
difference between this slow reversal of the magnetizing current and one
cycle per second than between the latter and merely reducing the current
from its maximiun value to zero without reversal.

9000

£900

8000

1500

1000

1000 8000 9000 4000 1

lUjnMlls p«r sq. em.

Fig. 2.

Permeability curves. I. After reduc-
ing the magnetizing field from ff » 60 to
H »0 by many reversals at one cycle per
second. II. After breaking the magne-
tizing current from H » 90. III. After
demagnetization by reversals once in ten
seconds.

Conclusions.

The proper magnetization curve for a given sample of iron is not ob-
tained unless the iron has been completely demagnetized. Reversals
of the magnetizing current at the rate of once a second may be too rapid
for effective demagnetization.

* Phys. Rev., Vol. 9, p. 415.

290

ARTHUR WHITMORE SMITH.

Sbcokd

When preparing to demagnetize a ring of iron, or other magnetic
circuit, the time required for the magnetic flux to become fully reversed
should be determined. This is readily done by closing the galvanometer
key after the reversal of the magnetizing current and noting what interval,
/, is necessary in order that there shall be no deflection. For complete
demagnetization the current should not be reversed faster than once in
2/ seconds.

This interval is not constant even for the same ring, but is longer in

the region of greater permeability. In this region, therefore, the current

should be reversed more slowly than at higher magnetizations. In every

case the reversals should be. slow enough to allow the flux to reach its

full value before the next reversal. This rule allows faster reversals at

the higher magnetizations.

Physical Laboratory,

University of Michigan,
May XX. 1917.

No"^'] ELECTRICAL CONDUCTIVITY OP SPUTTERED FILMS. 29 1

THE ELECTRICAL CONDUCTIVITY OF SPUTTERED

FILMS.

By Robert W. King.

TT has been known for a long time that the specific resistance of the very
A thinnest metal films is abnormally high.^ Two theories have been
advanced to explain this fact; one by J. J. Thomson* depends upon a
shortening of the mean free path of the conducting electrons by the surface
of the film; the other by Swann* depends upon an assumed granular
structure of the film and consequent opposition offered to the motion of
the electrons by the gaps between the grains.

Swann* has raised one objection to Thomson's theory. Another has
developed as a result of the present work. Thomson gives as the ex-
pression for the mean free path of an electron in a film

X' = /(f + ,-log^). /<X.

X being the mean free path for the metal in bulk, and i being the film
thickness. Evidently X' varies less rapidly than the first power of /;
and since the specific conductivity, other things being the same, is prob-
ably proportioned to X,' we find that the specific conductivity of a film
should vary less rapidly than the first power of /. This, however, is
certainly very seldom if ever the case. The writer finds this exponent,
instead of having a value less than unity, to have values ranging between
10 and 50, and sometimes reaching as high as 200.

Swann's theory, on the other hand, seems open to at least two objec-
tions. First, it is difficult to picture the mechanism by means of which
in all cases the grains of the film are distributed so as to lie separated from
one another by gaps of practically uniform width. The natural supposi-
tion would appear to be (as indeed Swann suggests) that of a more or less
random distribution, in which certain grains would undoubtedly touch
some of their neighbors and be quite distantly separated from others.
When two grains actually touch, it would seem reasonable to suppose

^ I. Stone, Phys. Rbv., 6, i, 1898. Vincent, Ann. de Chin, et de Fhys, (7), 19. 494, 1900.
Longden. Phys. Rev., ii, 40, 1900. Patterson, Phil. Mag. 4, 1902.

* J. J. Thomson, Cambr. Phil. Proc., 11, 120, 1901.

* W. F. G. Swann, Phil. Mag., 28, 467, 1914.

292 ROBERT W. KING. ^^S.

that the resistance of their contact is not abnormally high. Second, the
effect of such gaps between grains, as Swann imagines would, on the
whole, be to lengthen the mean free path of the electrons above the value
holding for those in the metal in bulk. This in turn should make the
resistance of the film more than usually susceptible to the action of a
transverse magnetic field. Such a susceptibility the writer has however
failed to find in films of either platinum, gold or silver.*
The following is an attempt to put the supposition of a random dis-
tribution* of grains into a quantitative form.
Imagine the points of Fig. i to represent the
centers of the metallic grains. The components
of all pairs of these points which lie less than
a certain distance apart have been joined. The
result is the formation of complicated net-like
paths over which it will be supposed conduction
can occur. The problem is to repress the total conductivity of these
paths as a function of the number of particles composing the iilm.

Let c represent the conductivity of the film, JV the total number of
grains per unit area, and n the smaller number through which conduction
can occur. Since the w grains which constitute the conducting paths
must, taking the film as a whole, be practically uniformly distributed,
we may suppose

n - cf{N).

where }{N) is a function which may be expected to decrease slightly in
value as the conducting paths become greater in number and consequently
straighter. Now consider the effect of increasing N by the addition dN.
The fraction c/(JV)/JV X dJVof these particles will on the average evidently
become conducting particles. But as a result of the addition, other
particles from among the N — n already upon the film will change over
to the conducting kind; and it probably is quite safe to assume that the
number which make this change is proportional to the number added
cf(N)IN X dN. That is, each particle which, when added, is of the
conducting kind, will on the average enable four or five or some other
number p of the N — n particles to change to the conducting class. This
number p must of course vary some with JV; particularly will this be so
as N gets so lai^e that the grains of the film begin to form more than one
layer, in which case it will tend toward zero. But we can doubtless say

■ It might be recalled here that Patterson (I. c) obtained similar resultB for platinum and
silver, and that for bismuth h« found the change of rcBistance to be noticeably less for a film
than for the metal Id bulk.

* It should be borne in mind that the distribution of grains cannot be random in the senM
that a distribution of point* might be random, since the grains have an appreciable alie.

fS!'^'] ELECTRICAL CONDUCTIVITY OF SPUTTERED FILMS. 293

that throughout a certain portion of the development of the film, p will
remain constant. Then, since

we have

dn^cdf+fdc^ (p+i)^iV,

dc df , . ^dN

log c + log/ = (p + i) log iV + const.

As pointed out above, the variation of /(iV) is likely to be slight in any
case. Let us suppose it constant. Then

log c =» (p + i) log iV + const.,

which is a relation agreeing very well with experiment.

Experimental Procedure.

The vacuum tube in which the sputtering was done presented no note-
worthy features except that it was provided with a pair of wires, leading
from the film to the exterior, by means of which the resistance of the film
could be measured in situ. Each film was approximately i X 1.5 cm.
and was deposited upon a piece of glass immediately after two dense
patches of film to serve as contacts had been deposited. Before beginning
the film, the discharge was run for a period varying from a few minutes
to half an hour, depending upon the metal used as cathode; and during
this preliminary discharge the plate of glass was protected by a cover of
glass. After conditions within the tube appeared to have become steady,
the discharge was stopped and the cover glass slid away by tilting the tube.

The deposition of the film was now carried out in small stages. In the
case of platinum and silver, the interval used was two seconds, while for
gold one second seemed more suitable. These periods of deposition were
accurately timed by means of a slowly falling piston, the piston being
released by the starting of the discharge, and it in turn stopping the
discharge upon reaching the bottom of its fall. The discharge was ob-
tained from an induction coil operating on an alternating current, one
half of each secondary wave being suppressed by a kenetron in series
with the coil and tube.

At the end of each interval of deposition, the conductivity of the film
was tested for or measured. The measurements were carried out by
noting the current which a known potential difference would send
through the film and a sensitive galvanometer placed in series. The
resistance of the galvanometer was enough smaller than that of the films

294

ROBERT W. KING.

to make correction for it unnecessary. The greatest resistance measured

was about 1.4 X 10" ohms.

The data of Table I. are typical of the values obtained for gold, silver

and platinum.

Table I.

Gold.

Silver.

Platinum.

Time.

Conductivity.

Time.

Conductivity.

Time.

Conductivity.

31 sec.

3.4XlO-wi

80 sec.

7.1 X 10-" 5

10 tec.

2.2XlO-"g

4.6X10-*

9.6 X 10-"

8.9 X 10-"

4.2XlO-«

1.8X10-*

12.2XlO-«

3.9X10-»

1.6X10-«

10.3XlO-«

2.9X10-*

1.6XlO-»

6.4X10-»

2.7 X 10-*

100

i.ixio-«

2.1X10-*

4.8 X 10-*

104

3.4X10-«

3.9X10-*

5.7 X 10-*

The accompsuiying curves, with the exception of Fig. 6 show to what
extent films of platinum, gold and silver agree with the relation given

Fig. 2.
Curves for platinum obtained by sputtering in a large bell-jar.

above.^ The films of platinum giving the curves of Fig. 2 were sputtered
in a large bell-jar having about three times the capacity of the vacuum
tube used in making all the other films. Otherwise conditions were as
nearly as could be judged the same.

It is evident from the curves that p stays constant for a greater range

^ In plotting these curves it has been assumed that N is proportional to the time of sput-
tering.

VOL.X.1

No. 3. i

ELECTRICAL CONDUCTIVITY OF SPUTTERED FILMS.

295

of values of the conductivity in the cases of platinum and gold than in
the case of silver. However, the deviations of the curves for silver may
be partly due to the difficulty experienced in keeping the conditions
within the tube constant when using a silver cathode.

Computing the range of values of N throughout which p remains con-

Fig. 3.

Curves for platinum obtained by sputtering in a small bell-jar.

stant brings to light peculiar differences. Table II. gives average values
of the ratio At/to, to being the time during which deposition must occur
in order that the film just show a measurable conductivity, and At the
time from the beginning of conduction to the end of the straight portion
of the curves. The corresponding values of p are also given.

Table II.

M«tia.

A///0.

Pt I

0.35
2.40
0.26
0.32

Pt II

" "0 •• .•...»•..»...

The variations of p and At/to among the various films are probably due
to different degrees of regularity of arrangement of the grains. If for
some reason the arrangement in a given film is very regular, it would be
likely to cause a large value of p and small value of At/to. This point as
well as a possible explanation of the difference between platinum I. and
II. will be returned to presently.

The lowest point on each of the curves for gold does not have much sig-
nificance. Gold films, when they first begin to conduct, seem to undergo

296

ROBERT W. KING.

a rapid growth of conductivity without the addition of any metal. For
example, the film from which Curve 5 was plotted, when first observed
after 31 seconds deposition, had a conductivity so small as to be scarcely
measurable. At the end of about a minute, this had increased over a
hundred-fold to the value plotted, and even then was increasing slowly.

Fig. 5.

Curves for silver.

This spontaneous increase, however, rarely if ever put in an appearance
after the next addition of metal. Neither platinum nor silver showed
ageing to a detectable extent, although it is likely they would have done
so had the time of observation been lengthened sufficiently.

No effort has been made to measure the actual thicknesses of the various
films. Certain limits can however be set with a fair degree of accuracy.
The films of platinum at the conclusion of deposition were still so thin
that, when looked at through the glass slide, they appeared as dark patches
on a bright background, this being due to the fact that the reflecting
power of the surface between glass and film was less than that between
glass and air. The gold and silver films were thick enough to make
their reflecting powers on the glass slide about equal to that of a clean
glass surface. From these facts, and making allowance roughly for the
variation of optical constants with film thickness,^ it is probably safe to
conclude that the final thicknesses lie between 6 and io/li/li, the platinum
films being somewhat the thinnest.

These values of the thickness give a rough indication of the average
size of the particles for the different metals. As mentioned above, gold

» W. Planck. Phys. Za., 15, 563, 1914. B. Pogany, Phys. Za.. 15. 688. 1914.

No^a!^'] ELECTRICAL CONDUCTIVITY OF SPUTTERED FILMS. 2gj

and silver films did not as a rule show any conductivity until about three
fourths of the final amount of metal had been deposited. Platinum
films (II.)» on the other hand, began to conduct after about one third of
the final amount of metal had been deposited. We may therefore say
that a platinum film, in order to just show conduction, must have an
average thickness between 1.5 and 3 /li/x, while gold and silver films have
an average thickness between 6 and 8 /n/x. Doubtless the average sizes
of the particles for the differet metals are considerably in excess of these
thicknesses.

Direct evidence as to the existence of these particles has been ob-
tained. Various investigators have examined films microscopically, but
with the exception of HouUevique^ have failed to detect any signs of
structure. Houllevique records that a film of silver about 10 /x/x thick
when examined with a magnification of about 1,300 diam. appeared con-
tinuous but granular. It occurred to the writer to try a "dark field "
microscoi>e. In this instrument, films of gold and silver, of such thick-
nesses that they would be on the straight portions of the curve, show an
unmistakable granular structure; for somewhat thicker films, the granu-
lar structure has almost if not entirely disappeared. On the other hand,
no detail of any kind could be observed in platinum films; but as was
indicated above, the size of the platinum grains is probably considerably
smaller than those of gold and silver, and they were quite likely without
the range of the microscope.

This instrument had a magnification of about 500 diam., a power too
low to make it feasible in any but a few cases to estimate the average
distance between particles. One of these was that of a gold film which
probably consisted of almost enough metal to enable it to conduct. The
average distance between particles seemed about 500 /*/x. Of course this
gives no indication of the actual size of the particles.

The question naturally arises as to how these particles are formed. It
certainly is not easy to conceive of them as being detached as units from
the cathode by the positive ion bombardment. It seems much more
reasonable to think of the cathode as losing particles of practically atomic
size, these uniting later to form the larger agregates.^

This view receives striking support through a comparison of the writer's
data with those recently published by Weber and Oosterhuis concerning
films produced by evaporation. Such films, it is known, are built up by
the condensation of atoms. These investigators find that a platinum film
in order to just conduct must be about 1.5 /x/x thick, and a silver film must

' L. Houllevique, Cr., 148, 1320, 1909.
* Cf. Longden and Houllevique, /. c.

298

ROBERT W. KING.

[

be about 6 /li/x thick. The almost exact agreement of these values with
those obtained by the writer can scarcely be considered accidental, but
would seem rather to be due to the films having been made by essentially
the same process.
Nor does the similarity between the two sets of data end here. Fig. 6

,.^

■ " ■ ■ /
/
$

4
1

Pt;

/
/

•

/
/

/ /

/
/
1
1

/ ^

/
/
/

•

1 1

IM THICKHIM

Fig. 6.

shows the result of plotting the values of Weber and Oosterhuis in ac-
cordance with the equation

log c = (p + i) log / + const.

The agreement in the cases of platinum and silver is very nearly as good
as that given by the writer's data, while the case of tungsten is indefinite.
These facts therefore make it seem likely that in sputtering, the metal
leaves the cathode in about the same condition as if it were evaporated
at high temperature, so far as the writer is aware, there are no facts to
which such a view runs counter.

Now since metallic atoms and small clusters of atoms display a marked
tendency, even at ordinary temperatures, to merge together when brought
sufficiently close, the particles found in sputtered films are readily ac-
counted for. But the process of condensation may occur at different
stages; entirely during the passage of the dark space, or entirely upon
reaching the glass, or partly in each place. The stage at which it occurs

Na"i^] ELECTRICAL CONDUCTIVITY OF SPUTTERED FILMS. 299

for auiy particular film doubtless depends upon the conditions of vacuum
and discharge as well as upon the metal used as cathode. And it seems
likely that the manner of growth of the conductivity of any particular
film would in part be determined by the condition of the metal at the
instant of deposition, — whether it is in fully formed aggregates, or in
atoms or small clusters which are to unite with one another to a greater
or less extent after striking the glass. In this way a possible explanation
is formed of why the values of p and A///0 differ so materially for the
platinum films of Figs, i and 2.

A satisfactory theory of film structure must account for the optical as
well as the electrical peculiarities. W. Planck and B. Pogamy (loc. cit.)
have recently measured the indices of refraction and coefficients of
absorption of sputtered platinum and copper films. They find certain
variations which they propose to account for by an assumed shortening
of the electronic mean free path. But it should be pointed out that their
values are quite similar to those Garrett^ has shown would be exi>ected
on the basis of- a granular structure, and are therefore in accord with the
theory here presented.

The question of a negative temi>erature coefficient has not yet been
investigated, but the writer considers it likely that this phenomenon will
find am adequate explanation in terms, partly of a differential expansion
between glass and film, and partly in terms of the remarkable tendency
to unite which minute particles of metal show even at ordinary temi>er-
atures.

In conclusion the writer wishes to thank Professor F. K. Richtmyer for
his help and constant interest. He also wishes to acknowledge assistance
from the Rumford Fund for the purchase of apparatus.

Summary.

1. Reasons are given for rejecting Thomson's and Swann's theories of
the abnormally small specific conductivity of metal films.

2. A relation connecting conductivity and thickness is deduced from
the supposition of a more or less random arrangement of groups of atoms.
This relation seems to fit observations upon films of platinum, gold and
silver in a very satisfactory manner.

3. It has been found that the thinnest films of gold and silver show a
granular structure when examined with a ** dark field " microscope, and
that. thicker films of these metals appear quite uniform. No structure
has been observed in platinum films, but this is probably due to the
limitations of the microscope.

» J. C. M. Garrett, Phil. Trans., A, Vol. 202.

3CK) ROBERT W. KING. [

4. It has been found that in order to just conduct, platinum films must
be between 1.5 /x/x and 3 /x/x thick, gold and silver films between 6 /x/x and
8 /x/x thick. As it seems doubtful if particles of the sizes necessitated by
these thicknesses can be detached from the cathode by the bombardment,
their formation is probably due to the condensation of atoms of the metals.
This supposition is further supported by the similarity between the
writer's results and those of Weber and Oosterhuis obtained on films
produced by condensation.

5. It is pointed out that the recorded variations of n and k with thick-
ness appear to present no obstacles to the acceptance of the present
theory.

Cornell University.
June I. 1917.

Vol. X.!
Na3. J

THE MERCURY-ARC PUMP,

301

THE MERCURY-ARC PUMP; THE DEPENDENCE OF ITS RATE

OF EXHAUSTION ON CURRENT.

By L. T. Jones and H. O. Russell.

OINCE the diffusion pump of Gaede^ was described a number of in-
^ vestigators have produced modifications all operating on the same
principle, the latest that of Knipp.* The pump here described differs
only in the manner of driving and in introducing and taking off the mer-
cury. This permits using the pump as a mercury still at the same time
that it is being used for exhaustion purposes.

Fig. I is a reproduction of the pump in its most desirable form. Two
barometer columns introduce the mercury to the arc, the arc being
started by blowing in the one neck of the Woulff bottle as shown at B,

■J

•

rig. 2.

'-r 1/

ompa. Ao

Fig. 1.

Fig. 2.

The mercury vapor is driven through the nozzle, N^ and condenses in the
chamber surrounded by the water jacket, J. The condensed clean mer-
cury is then drawn off at 0. The water jacket is conveniently made of
metal and the ends made watertight by rubber stoppers. Danger of
breakage, when made of ordinary glass, was encountered only at the
higher current values, 15-30 amps., which heat the arc quite above the
temi>erature necessary for highest efficiency. With the pump constructed
of Pyrex glass no difficulty is experienced due to breakage.

» Ann. d. Phys., 46, p. 357, 1915.

* Phys. Rev., N.S., IX., p. 311, 1917.

302 L. T. JONES AND H. 0. RUSSELL. ^2S

The supporting punp was a Gaede rotary mercury pump connected to

P. The vessel to be exhausted was connected through a liquid mercury

stopcock to the intake, I. Fig. 2 shows the relation between the driving

current and the rate of flow of the mercury vapor through the nozzle.

Each point represents a value deduced from a one-hour run at that

current value. The curve is evidently a straight line for values above

3 amps. The dotted line gives the

most probable form of the curve

for lower values.

The pump was connected
through about 200 cm. of 2 cm.
tubing (liquid mercury valve but
no vapor trap) to a 6.5 liter vessel.
The pressures were measured by
means of a 500 c.c. McLeod gage
and the rate of exhaustion was de-
termined by pumping always be-
tween pressures for which the Mc-
Leod gage is reliable. The annular
space about the nozzle, N (Fig. i,)
Pig 3_ was lirst made of 1.5 mm. width.

The one curve of Fig. 3 shows the
rate of exhaustion for this annular opening as calculated by Gaede's
formula*

where pi and pt are the pressures before and after exhausting the volume
V for / seconds. The speed, S, is in c.c. per sec.

To determine the reason for the maximum rate of exhaustion the water
jacket was removed and the annular opening blown out to 4 mm. width.
Fig. 3 shows that the rate of exhaustion was not markedly altered. The
limiting rate of 400 c.c. per sec. is due to the quite long length of 2 cm.
intake tubing. If this tube were short enough and large enough the curve
would most likely be a straight line after passing the 3.5 amp. value until
the limiting value due to the size of the annular opening is reached. This
suggested that if the pump be allowed to operate without reducing the
pressure, as in pumping against a leak, a definite number of mercury
molecules are required to remove a single air molecule.

Fig. 4 shows the relation between the number of mercury molecules
per air molecule and the pressure (average of pi and pi) in cm. The

'Loc. cit.

VoL.X.1

THE MERCURY-ARC PUMP.

303

curve is plotted for the current value 6.15 amps. Curves were plotted
for other current values and found to be not essentially different. Table
I. gives the data from which the curve is plotted.
The curve is not unexpectedly asymptotic to the vertical axis since

ons^

Pressure in cm*
Fig. 4.

isSS^

mercury vapor flows through whether air remains to be exhausted or not.
The fact that at the higher pressures only 1,000 mercury molecules are

Table I. *

Tim« of

Pumping,

Sec.

/,Cm.

/tCm.

Speed of

BzOAUStiOD

Cc. per Sec.

Molecules of Air
per Sec.

Molecules of

Mercury per

Air Molecule.

20.7

.00166

.00064

300

2.50X10»»

1.220

21.2

.00064

.00017

413

1.15 "

2,650

21.0

.00017

.00005

357

.28 "

11.100

20.0

.00178

.00069

306

2.80 "

1.090

20.4

.00069

.00018

443

1.29 "

2.360

20.4

.00018

.00006

380

.31 "

9,700

20.2

.00152

.00047

376

2.67 "

1.140

20.3

.00047

.00013

370

.86 "

3.540

20.0

.00163

.00052

370

2.85 "

1.070

19.8

.00052

.00014

425

.98 "

3.130

required to direct the motion of an air molecule is particularly interesting.
It remains to be seen whether this unexpectedly small indicated value
holds for pressures as high as atmospheric. The straightness of the path
of an alpha particle passing through saturated air at atmospheric pressure,
as is so nicely shown by C. T. R. Wilson's^ well-known photographs, is
interpreted as intermolecular penetration. The relative velocity of the

* Phil. Trans. Roy. Soc., A, 189, p. 265; 192, p. 403; 193, p. 289.

304 L, T, JONES AND H. O. RUSSELL. [^^

mercury and air molecules is quite small compared with that of an alpha

particle and the time they are within a given distance of each other is

correspondingly long. Whether this fact is sufficient to account for the

small number of mercury molecules required to direct an air molecule or

whether there is an essential difference between the mercury and helium

atoms cannot at present be told. The authors had hoped to continue

the work by several obvious experiments but other duties will surely

prevent it.

Physical Laboratory,

University of California,
May I, X917.

Second Series. October, igiy Vol. X., No. 4.

THE

PHYSICAL REVIEW.

KINETIC THEORY OF RIGID MOLECULES.

By Yoshio Ishida.

Introduction.

T was shown by Boltzmann^ that the behavior of a monatomic gas
may be studied by means of the partial differential equation

when the gas is an ensemble of the same kind of monatomic molecules.
The/ is the number of molecules i>er unit cell; the op, y, «; f, ?;, f ; X, F, Z
are components of three dimensional space, velocity, and acceleration
respectively and the J is the rate of change due to encounters. The
obvious extension to any number of dimensions is

dr^ dxi "^^

where the jc's are any coordinates to specify the system of the individual
molecule.

Now the deductions from this equation may be classified into two cate-
gories; namely, those which are independent of the form of 7, and those
which depend upon the nature of /. The hydrodynamic equations can
be derived without knowledge of /, provided we admit the existence of
such a function. On the other hand the quantitative determinations
of the pressure, the viscosity, and the thermoconductivity can not be
effected, unless we know something about /.

Since the form of / depends upon what is assumed concerning the
nature and frequency of various types of encounters between the molecules
it is convenient to classify the coordinates into two groups, according as
they are or are not affected by encounters. Let us call the first, the

1 Boltzmann, Gas Theorie, Vol. I., § i6.

305

306 YOSHIO ISHIDA. to»

affected codrdinates, and the latter the immune codrdinates. During
encounters, if there is a function of the affected codrdinates such that the
sum of the function of the coordinates for one molecule and the same
function of the coordinates for the other molecule remains unchanged, we
shall call such a function an invariant of the encounter. Confining at-
tention to binary encounters, if we have k affected coordinates of one
molecule, then the question is to determine 2k variables after encounters
in terms of 2k variables before encounter. If. there are r invariants in
this special sense in addition to the one purely numerical invarisuit, and
s other general relations (without arbitrary parameters), then the equa-
tions of encounter will involve 2k — {r + s) parameters. Let *< be
the invariants of encounters including ^o "^ if then the equations

dcij^iJdcj^ = o; i = o • • • r; will be valid and will give f + i funda-
mental equations of what may be called generalized hydrodynamics
corresponding to the space of the immune coordinates. The d^i is an
element of the immune space and the d<r^ is that of the affected space.

To illustrate the notion, let us consider the case of a monatomic gas.^
The jc, y, z are immune coordinates, and {, ?;, f are affected coordinates.
The number, the three components of translational momentum, and the
energy of the system are invariants of encounters, so that r = 4. For
the parameters of an encounter, we have the longitude and latitude of the
point of contact if we adopt the idea of an elastic sphere; the distance from
the asymptotic line and the orienting angle if we choose the conception
of central forces. Therefore 5 = 0, and consequently there are no ad-
ditional relations entering into the consideration, and also there are no
more invariants. The conservation of the number gives

jJdcj^ = o (the numerical invariant),
where (/<r^ = d^dridj^, which reduces to the equation of continuity

^+V.(pV)=o,

where p = fmfda^, and V(F„ F„ V.) = W(f, rj, f) - U(C7., C7„ U.).^

The V is the mass velocity and U is the velocity of agitation.
The conservation of translational momentum gives

J wW7d<r^ = o,

1 Boltzmann, Gas Theorie. Vol. I. Maxwell, Collected Works. Vols. I. and II. Lorentx,
Collected Works, Vol. II. Kirchhoflf. Theoretische Physik, Vol. IV. Hilbert. Math. Annalen,
Band 7a, 1912. p. 562.

' In order to save space. Gibb's vector notation is used in this paper whenever it is con-
venient. The vectors are indicated by clarendon type.

NoI"^*l KINETIC THEORY OF RIGID MOLECULES, 307

which, when combined with the equation of continuity, reduces to the
equation of motion in hydrodynamics,

{^ + V. vv} + V-A-pg = o,

where A is the dyadic of stress and g is the vector of acceleration. Finally
the conservation of energy gives

m — 7d(r^ = o,

2
which reduces, when combined with two preceding, to

p{^+V • Ve} + V-h- V • (A • V) - V- (V- A) =0,

where h is the flux vector of Fourier conduction of heat and e is the thermal
energy per unit mass.

The present paper considers some features of the kinetic theory of
gases under the assumption that the molecules are rigid bodies, having
no spherical symmetry. The first part will deal with the general
hydrodynamical relations. It is evident that we have to consider the
orientation and the angular velocity^ of each individual molecule besides
its space coordinates and translational velocity. For invariants, we
have three additional equations stating the conservation of moment of
momentum. Then the space co5rdinates x, y, z and the angles ^, ^, 9
are regarded as immune coordinates so that the general idea explained
above leads in the first place to a kind of hydrodynamics of six dimensions.
The three angles of orientation are then integrated out so as to leave the
suitably modified equations of ordinary hydrodynamics, together with
an additional vector equation corresponding to the conservation of
moment of momentum, which suggests the possibility of the propagation
of gyroscopic disturbances besides the sound waves. It will be shown
also in the second part that we can specify such binary encounters
by five parameters. Consequently we have twelve variables, with seven
invariants and five parameters of encounters, thus forming a complete
system in the sense that all the independent invariants have been utilized.

The investigations of the specific heat of gases^ from the standpoint of
the equipartion of energy indicate that we can not treat gases like oxygen
or hydrogen as monatomic. Thus we have to consider the energy of
rotation, which is caused by asymmetry of shape and loading. The idea

1 Tiflflerand, Mte. Caeste, Vol. II. Poisson* M^.» Vol. II. Appell, Mte. Rationelle,
Vol. III.

> Kirchhoff, Theoretische Physik, Vol. IV., page 169. Jeans, Djmamical Theory of Gases,
pages 8z and 171. Raleigh, Theory of Sound, Vol. II.. page 18.

3o8

YOSBIO I SB IDA.

[

of considering gas molecules as rigid bodies was initiated by Maxwell,^
who computed the impulse if two such bodies were to collide. Later
various writers* carried out the work for some 8p>ecial cases. The second
part of this paper will deal with a collision axiom for a more general
type of rigid bodies, and some of its consequences. We shall discuss the
distribution of translational and angular velocities, especially the equi-
librium distribution and its relation to the jff-theorem. The distribution
function thus deduced will be utilized to compute the external pressures

of such gas molecules.

I. Hydrodynamics.

Let ^, y, 2; {, i;» r be the translational space and velocity coordinates.

For orientational coordinates, we can use the Euler angles ^, ^, ^; and

angular velocities «i, «2, «i (see Figs, i and 2). The system of moving

Fig. 1.

Fig. 2.

axes {xu yu ^i) and fixed axes (x, y, z) are connected by the following
equations (if the translational motion is temporarily negleeted).

X = lxi + tnyi + nzi

y = Vxi + tn'yi + n'zi

z = V'xi + fn"yi + n"2i,

the nine direction cosines being expressed in terms of the Euler angles,
as indicated in the following schema

cos ip cos \^ — sin ^ sin \^ cos ^,

— cos ^ sin \^ — sin ip cos \^ cos d, sin B sin tp

sin ip cos 4/ + cos tp sin \^ cos ^,

— sin ^ sin ^ + cos ip cos ^ cos d, — sin B cos ^

sin \^ sin ^, cos \^ sin ^, cos 6

» Maxwell: Collected Works, Vol. I., page 406.

« Jeans, Dynamical Theory of Gases, p. 93. Burbury, Phil. Trans., A CLXXXIII.. p. 407,
1892. Bumside, Trans. R. S. E., XXXIII., part 11.. 1887. N. Delone, Report of Russian
Imp. University. 1892.

/, m, n

:=4.

NoV^*] KINETIC THEORY OF RIGID MOLECULES. 309

Then the three components of the angular velocities may be expressed in
terms of the time derivatives^ of the angles,

«i = ^ sin B sin ^ + d cos \^,

«2 = ^ sin B cos ^ — d sin \^,

«8 = i? cos B + '^.

The auxiliary formulae for the change of direction cosines may be obtained

directly, thus •

• • •

m = nwi — /«t, w' = n'«i — /'ws, m" = n"wi — /"wti
n = /w2 — w«i, n' = V<ji% — m'wi, n" = /"«2 — m"«i.

If the moving axes are chosen as the principal axes of the body, the dyadic

of inertia is

T ^ Aii + Bjj + Ckk,

where At B^ C are the principal moments of inertia, and t, j, k are unit
vectors along the moving axes.

Then the differential equations of the motion of a molecule are

i = f , y ^ Vf « = r,

and

sm 4/ cos 4/

^ sm ^ sm B '

1^ = «8 — cot ^(sin ^wi + cos \^W2),
d = cos ^«i — sin \^«2,

5W2 = (C ^ i4)wiW8 + -Sf,

Cci, = (.4 - B) 0)10)2 + N,

where X, F, Z are the components of the impressed force, and L, Jlf , iV
are the components of the impressed couple. The first specify the motion
of the center of gravity, and the second specify the rotation of the body
referred to the principal axes.

If / is the number of molecules per unit cell in the twelve dimensional
region, and / is the rate of the change of this number of molecules due to
encounters the Boltzmann equation may be written

dt^ dx ^ d^ ^ dip ^ dcji '

> The molecular time derivative is designated by placing a dot above the character, where-
as the molar time derivative is designated by the ordinary form d/di, d/dL

3 I O YOSHIO ISHIDA .

Let us put for brevity

dT = docdydz, dr' = dipdy^dB^
dff = d^Tfd^f dff' = d<a\d(aii(at.
The range of the variables will be

i n i \ ip yp B; (jii (at «t .

— 00 — 00 — 00 o OO — 00 — 00 — 00

+ 00+* +* 2T2irir +00 +00 +00

First let us deduce hydrodynamical relations in six dimensions. The
conservation of the number gives

J Jdada' = o.
Now define the density by the equation

p* = J mfdtrda'

so that

dp* C df

and also introduce the notation W = U + V, where W is the velocity
vector of the center of gravity of the molecule in question, its components
being {, iy, f ; V is the vector of mass velocity, and U the vector of agi-
tation velocity. It follows that j mJSfdada' = o. Then we have

Jjdcdc' = ^ + V (p*V) + J f^ I Xxjfdridl: } da'

+ I mS — — dadff' + I *w I 2«i I /dwirfwf f d<r.

Further let us specify the nature of the external forces and torques,
and the distribution function/ in the following manner:
(i) Suppose X, F, Z are independent of the velocities.

(2) Suppose L, My N are independent of the angular velocities.

(3) Assume / to be such a function with respect to {'s and co's, that the

surface integrals become zero as the surfaces extend to infinity.
Then the third and the fifth terms reduce to zero. Let us now write the
fourth term as follows

mX^—d<rd<r' - V* • (p*N*),

0(p

where

/ d d ^ \

NoI"^l KINETIC THEORY OF RIGID MOLECULES. 3 1 I

and

p*N* = zfmfixliTdir'.

Also N* is expressible as a linear combination of M*, where M* is the
vector of moment of momentum in the fixed space,

Ml* = («"i4 + mm''B + nW'QNi* + nCNt* + cos ipNi*,
Mt* = (ITA + mWB + n'n"C)Ni* + n'CNt* + sin ipN,*, ]

Mm* = (V'^A + m'^B + n^^QNi* + n"CNt* + o.

The symbol V^ may be called angular divergence following the analogy
of ordinary space. So finally we have

^ + V • (p*V) + V^ • (p*N*) - o,

which is the equation of continuity in six dimensions.

The conservation of translational momentum, namely jmSRJdada'
may now be considered. If we define the dyadic of stress by

/ mUXJfdffdff' = A*,
we shall have

fmWW ' Vfdadir' = V • {p*W + A*}.
Also we have

where

/«w|d.d.'=|(p*V).

fmg • V^fdffdff' = o,
fmWg • V^fdada' = - p*g*,

d d d

With this notation we have

mSRJd<rd<r' = ^^^ + V • (p*W + A*) - p*g* + V^(p*N*)V = o.

Finally

p*^ + V . A* - p*g* + W^ • (p*N*) = o.

The conservation of energy may be treated in a similar manner.
The energy of translation for the molecule in question is

E = \m{^ + 1;* + r*).

312 YOSHIO ISHIDA, \\

and for the rotational energy we have

so that the conservation of the total energy gives

/ (£ + K)Jd(fda' = o.
Let us adopt the notation

hffnTPfdada' = p%*,

where e«* is the thermal energy per unit mass due to the translational
velocity,

fKfdadff' = p*er*
where Cr* is the thermal energy per unit mass due to the angular velocity;

e* = Ct* + er*;

where h* is thermal current density corresponding to the Fourier con-
duction of heat;

/ (iwU2 + K)2i>fdada' = S*,

where S* is the energy flux carried by the angular velocity;

ffimL + o)tM + u)zN)dady = g*,

where q* is the work done by the impressed torque. We further have the
following reductions:

i / mW^fdad^ = ip*V2 + p*«i*,

ifmW^Wfdada' = V(Jp*V* + p*et*) + V • A* + h*.
With these auxiliary formulae we obtain the energy equation

I iy*^ + P*e*) + V • {(ip*V^ + P*e*)V + V • A* + h*l

+ V^ • {ip^V^N + S*} - V . (p*g*) - 2* = o,
which reduces to

P*%+ p*V • Ve- + V • (A* • V) + V • (V • A*) + V h*

- 2* + V^ • {ip*V«N* + S*l = o.
The conservation of angular momentum gives the equations

f XlAuiJdadff' = o,

No"^'] KINETIC THEORY OF RIGID MOLECULES. 313

f Hl'AcoiJdffdff' = O,

f iV'AcoiJdadff' = o.
Let us work out the first component. We have

XlAwiT-.dadc' =

at at

fxlAu>iV • UW)dada' = V • (p*VMi*),

J 2/i4«iV^ • (Jg)dada^ = o.

If we write

ff<p2lAo)idada' = Hi*,

where Hi* is expressible as a linear combination of three components of
the rotational energy, we have

r XIA coiX ^^ dada' + f XlA «iS ^^^ dcrdcr'

= V^ • -ffi — J f(lA<ai + m5a>j + nC(az)dad(r\

The integral reduces further on account of the Euler equations and the
equations of the change of the direction cosines, to

- ff{lL + mM + nN)dcdc' = - Gi*.

By symmetry we obtain the second and third components, so that we
have

^\^ ^ + V • (p*VM*) + V^ • {H*) - G* = o.

Thus we have deduced a complete set of a kind of hydrodynamic equa-
tions for six dimensions.

We can, however, further integrate out the angles of orientation and
obtain the resulting system of equations in three dimensions. Since the
frame of reference for the Euler angles is arbitrary, the condition that
/ is a continuous function of ^, ^, and d implies

[/]$:5' = o, [/]$:?' =0, [^-r

= o.

sm 0

The space of integration dcr^ is now dr'dada' instead of dad<r\ and we
have to redefine our notations in the following fashion,

P = J mfdT'dadff', etc.
It will be seen that, by carrying out the integrations, we have,

3 1 4 YOSHIO I SHI DA .

for the conservation of number,

^ + V-(pV) -o;
for the conservation of translational momentum.

^^+V-(pW + A)-pg = o;
for the conservation of energy,

|(ipV« + pe) + V • {(ipV» + p^)V + V . A + h} - 2 = o;
for the conservation of moment of momentum,

a(pM)

+ V • (pVM) - G = o.

We notice at once that these equations are exactly the same as the
preceding set provided we assume the angular divergences V^ to be zero.
We can further simplify the result if we use the Lagrangian time deri-
vatives^ instead of the Eulerian time derivatives. Thus

P^ + V-A-pg=o,

p ^ + V • h + V • (A • V) + V • (V • A) - g = o,

dU ^

"IT-®"**'

The first two equations are the same as for the monatomic gas. But
the third equation contains the rotational energy as well as the trans-
lational, and there is also a contribution of energy due to the work done
by the impressed couple. The last equation is the new statement, which
suggests that a gas consisting of nonspherical rigid molecules could pro-
pagate a kind of gyroscopic disturbance along with compressional waves
of the familiar type.

II. Collision Axiom and the Distribution of Velocities.

In the preceding discussion we defined / as the number of molecules
per unit cell. This function / will then depend upon thirteen variables

d d

1 -- « — 4- V • V
dt dt^^ ^'

iSo"^] KINETIC THEORY OF RIGID MOLECULES. 3I5

including the time, and our problem is to find these relations. This
ftmction / we shall call the distribution function, and assume to be a
continuous function with respect to all these variables. Let us call

p = J mfdr'dtTdij' the mass density.^ The molecular density (say v)

may be absorbed in /, so that we can keep the uniformity of notation.
At a given time we can classify all molecules according to twelve proper-
ties; then the number of molecules in one of the twelve dimensional cells
is

fdTdr'dadc' ,

We shall now consider the impact of two molecules which behave like
rigid bodies. Let Oi and 0% be the two cen- ^
ters of gravity, P the point of impact, and '
RiPRt the line of impact (normal to the N. -q

common tangent plane at P) (see Fig. 3).
Let the position of P with respect to the
principal axes through Oi be fi, and the same
with respect to those through Ot be fj. Let a
unit vector along the line of impact with re- ^

spect to Oi system be ai, and the same with Fig. 3.

respect to 0% system be a^. Take for the

moments of inertia along the principal axes in these two sets Au Bi,
Ci; -42, B2, Ci using dyadic notation, then

Ti = Aiiiii + BJiji + Cikiku

Fa = Aiiiii + B%j%J2 + CJiikt.

Take for the mass of the first body mi, and the second ntt. Let further
the translational and the angular velocities of the two bodies before and
after impact be

Wi,W2,Oi,02 and Wi,W2,Oi,02,

respectively. If we take R for the measure of the impulse due to the

impact, we have the following relations;

for the conservation of translational momentum,

miWl = miWi + Eli?,

W2W2 = m2W2 — 9LtR\
for the conservation of moment of momentum,

Ti . Oi = Ti • Oi + (rx X ai)i?.

Fa • O2 = r, • O2 - (r2 X 9Li)R\

3i6

YOSHIO ISHIDA.

rSscoND

for the conservation of energy,

yWi^ + yW^^ + iOi- ri-Oi + iO,. r,.o.

From this last equation we can obtain R in terms of the r's and a's, sub-'
stituting the values of the variables before impact for those after impact.
Thus

^ ai ' Wi - aa ' W. + (ri X aQ ■ Oi - (u X a^) - O2

:^ + :^ + [(r X ai) • Tr' • (rx X ax)I + [(r, X a,) • r,"^ • (r, X a^)]

If moreover the two molecules are of the same kind, Wx = mt equal to
m say, and Tx = Fa = F, then

^ ^ _ ax ' (Wi + Ox X ri) ~ a> - (Wa + 0« X u)

^^ I + m[(rx X aO • F . (rx X ax) + (ra X a,) • F • (ra X a,)] '

If we call the direction of the impulse the normal direction (normal to
the surfaces), the normal component of the relative velocity of the point
of impact will be given by

W, = ax • (Wx - rx X Ox) - aa • (Wa - r, X 0,).

We are now ready to consider the probability of impact of two such
molecules. Let us fix our attention only on these two molecules
which are going to collide. They will have rotation as well as motion
of the center of gravity, and it is necessary for us to observe not
only the motion of the centers of gravity but also the behavior of
the two points which are going to collide. Let the point of impact
of the first body be P and that of the second body be P'. Then if
we imagine the first body at rest, P' will describe a curved path
before it impinges on P, with such a relative velocity that its normal

component may be represented by Wn.
c ^0 Such a path may be found from the
differential equations of the motion if we
know Xf F, Z and L, M , N,

As a natural extension of the ordinary
supposition in the case of rigid spheres,
we shall assume that the probability is
proportional to the volume of a cylinder
whose base is the element of surface ds at P and whose slant height is the
relative velocity of the points of impact.

What we are required to find is, a pair of translational velocities and

Fig. 4.

Na'4^'] KINETIC THEORY OF RIGID MOLECULES, 3I 7

a pair of angular velocities after impact in terms of those before impact
and the parameters which specify the particular type of impact. One
formulation is to take two parameters to specify the point of tangency
on the first body and to take the remaining three to specify the orien-
tation of the second body with respect to the first body (see Fig 4).
Let Fi(ri) = o and Fi(ri) = o be the two surfaces; then the condition of
tangency will give

dFi ^ idFt. , dFi ,^Ft\
dxi \dxt ayt 0X2 /

. a^ = ^te^» +6^^^' +ai7'*^ r

where

X = ±

If the normal is taken in the sense of VF, the negative sign is taken.
Now the set of the direction cosines h, Wj, nj, etc., may be given by three
orienting angles say *, ^, 0. Then two parameters on the first body,
say the longitude and the latitude, will determine SFi/dxu dF/dyu
dFi/dzu and consequently dF^/dx, dF^dy%, dF^jdz^ and 12 may be obtained
as functions of these five parameters. Let us designate the element of
parametric space (with a proper proportionality factor )by dp\ then the
probability of impact is \7ff\dpdad(r\ Following the usual method^ let us
conceive two classes of molecules say A and B which are both distributed
in the element drdr' of space at random. We may suppose the transla-
tional velocities and moments of momentum to be uniform so that changes
occur only at a collision. Let us classify the encounters into two types
a and jS, where a designates such encounters that before the collision one
of the colliding molecules belongs to the class A and the other to the class
5, whereas jS designates such encounters that after the collision one of
the colliding molecules belongs to the class A and the other to the class
5, both types having the same line of impact (the common normal) and
the same orientations. The number of collisions of a type in unit time
per unit cell of r and r' space will be

f f\SfTn\dpd(Tidc^ai'd(Tt\

> For instance see Jean's "The Dynamical Theory of Gases," Chap. II.

3 1 8 YOSHIO ISHIDA .

where daidai refer to the class A and da^trt to the class B; and / and
/' are the distribution functions with arguments having subscripts i and
2 respectively. Then the total contribution to the class A due to this a
type will be given by integrating the above expression over all possible
0*2 and (Tt, namely

d<Tid<n'ffff\Wn\dpdc^c2'.

The number of collisions of type fi in unit time per unit cell of r and /
space will be

JJ'\Wn\dpdffidffJffi'dffi',

where the dashes above the characters express the corresponding functions
for the type jS, and the total contribution for the class A due to this P
type will be, then,

d^id^iffjJ'\Wn\dpd^^t\

In this theory we assume central symmetry so that dp = dp. Therefore
the number of moleailes in the class A is increased by the difference of
the two integral expressions above. The difference may be written in
the form

daidai'ff (J J' - //') \Wn\dpd<r^/-

This involves the fact that the Jacobian^ of the transformation is equal
to unity and \Wn\ = |W,»|.«

This is the expression for / from this point of view. Thus we for-
mulate the Boltzmann equation as follows:

•j-dffidffi = d<r}d<T\ '
at

dt^ dx ^ dk ^ dip ^ d(a \

= daidai' f(JJ' - ff)\Wn\daf4irt'dp.

Let us define

5 = - kff log / daidai,

where 5 = — kH, H being Boltzmann's probability function. We
obtain in the familiar way

Ti " i*/(^^8://' - log//0(7/' - ff')\Wn\da,d<r^tf,'d<rt'dp,

showing that dSjdt is always positive or zero, and 5 is an increasing
function or else constant. For the steady state 5 is a maximum and
therefore dSldt = o, so that we have //' — //' = o. This functional

^ It may be computed easily from the equation of the transformation to be — z. but since
we are concerned only with the numerical value the positive sign is taken.
* See Maxwell, Collected Works. Vol. I., p. 407.

VOL.X.1

KINETIC THEORY OF RIGID MOLECULES,

319

equation is equivalent to

log7+Iog7' = log/ + Iog/',

which is the form of an invariant of the encounters. Therefore

log / « an invariant,

is a solution, and the complete solution is a linear combination of all
invariants. Thus

log/ = aiN + a2(mV« + 0-r-0)+b-V + C:r.O,

where au ai; b and c are arbitrary constants. Taking the logarithm and
rearranging the expression, we have for the distribution function

Q . r

f^eie

•'[(V-Vo)'

(O-Oo)],

and the constants^ ^, or, Vo, Qo are to be determined by the total number,
the temperature and the visible motions of translation and rotation.

External Pressure for State of Equilibrium.

We have already found an expression for the impulse, when two rigid
bodies impinge on each other. In case of the external pressure, we can
simplify the expression, for we can take the plane of the wall as the x^-y
plane and the axis of s as the direction of the impulse. Thus

a . (W + 0 X r)

jR = — 2fn

I + m(r X a) • r • (r X a) '

where a has now for its three components V\ m", n". It must be noticed
that all the vectors in the above expression are referred to the principal
axes of the body. The distribution of the
orientation being the same as the distribution
of the point of tangency o of the x-y plane,
we may take the probability of impact to be
the product of the normal component of the
velocity of the point of contact and the prob-
ability of distribution of the 2-axis with re-
spect to the center of gravity (see Fig. 5).
This latter is given by 1/4T sin 6 dddfi, where
M is the longitude and $ is the latitude of s
on the unit sphere referred to the principal axes. From the geometry
of figure we can identify this 6 with the previous $ and /« with ^ -|- x/c.

1 These constant* may involve x, y, %\ ^, ^, $\ t.

Fig. 5.

320 YOSHIO ISHIDA. [

Therefore the probability of impact is

|Wn| — sin eded}^.

Taking the half of this probability because of the assumed central sym-
metry, we get for the pressure on the x-y plane

1 r r r' f fR\Wn\d<rd<r' sin eded^.
where

and

2? =— 2W

^+^1 A + B + C J

in co5rdinate expression. Since

/" = sin 4/ sin $, fn" = cos }[/ sin 6, n" =» cos 6,
we have

r— = X sm ^ sm ^, r-~ = X cos \^ sin ^, i— = X cos $,
axi ay\ oz^

and consequently if we know F, we can solve for jc, 3^, z as functions of ^
and B. We found above the distribution function /, and since the <r
and cr' spaces are independent of the form of F and the orienting angles,
we can at once effect the da and d<r' integrations.

If we assume the mass motion and the visible rotation zero, the ex-
pression for/ may be written

where €1 may be determined by integrating this expression over the whole

space, namely

+00

— 00

givmg

In the expression for the pressure, carrying out the integrations with
respect to dada'

p ^ —- \ I o^ sin ^ dedy^,
oTT Jq Jo

Na"4^'] KINETIC THEORY OF RIGID MOLECULES, 32 I

and if a is independent of the angle

Putting

p r= Nm — .

— S3 kT

2 *^'

p = NmkT,

giving Boyle-Charles's law for this kind of gas.

The writer wishes to express his gratitude to Professor A. C. Lunn,

who has given suggestions in carrying out this work.

Rybrson Physical Laboratory,
University of Chicago,
February 2. 19x7.

322

THOMAS E. DOUBT.

fSSCONO

^jsaEs^

TALBOT'S BANDS AND THE RESOLVING POWER OF

SPECTROSCOPES.

By Thomas E. Doubt.

WHEN a pure spectrum formed by a prism is observed visually with
one half of the aperture covered with a very thin piece of glass,
dark bands may be seen throughout the length of the spectrum crossing
it at regular intervals parallel to the slit.^* These bands are visible pnly
when the retarding plate is introduced on the side that the blue appears.*

The retardation of one half of
the beam by the interposed plate
may be accomplished in a num-
ber of ways.* Fig. i shows how
Talbot's bands may be produced
by means of the Bunsen-Kirch-
hoff spectroscope of single prism.
The retarding plate must be in-
troduced at i, i' or i". In Fig.
2, is shown the method of pro-
ducing Talbot's bands by means
of the Hilger wave-length con-
stant deviation spectroscope."
In Fig. 3 is shown the plan for
the most convenient arrange-
ment with a Rowland concave
grating." The retarding plate is
placed at P, the grating G and the
eyepiece of camera at 0. Fig. 4
shows the plan of arrangement
of the echelon used for producing these bands.^^ A is the light source, 5
the horizontal slit, C the collimator, E the echelon followed by a large
flint glass prism, P the plate which is thrust half way into the beam and
the telescope T, with the eyepiece or camera K. There were thirty ele-
ments, each about i cm. thick, in the echelon. Fig. 5 gives the plan used

» These numbers refer to corresponding numbers in the bibliography near the end of the
paper.

Vol. X.1
No. 4. J

TALBOTS BANDS.

323

with one of Michelson's gratings mounted Littrow. 5 is the slit, L
the lens of about 20 ft. focal length (6 m.)» G the plane grating, P the
plane parallel plate, and O the observer. Fig. 6 is a print of an exposure
made with the Hilger wave-length spectroscope. The source of light
was the carbon arc. This is the entire visible spectrum with the red end

toward the left. The retarding plate was about i mm. in thickness and
was placed at *, Fig.- 2.

Fig. 7 is a spectrum in the second order with a small Rowland concave
grating with a radius of curvature of about 180 cm. It is in a region
near the D line of the solar spectrum. The plate was plane parallel'
about 7 mm. thick and was placed as in Fig. 3. There are 7 Talbot
bands between the Fraunhofer lines Di and D^ which appear about one
fourth of the way across from the left end. Fig. 8 shows about 31
Talbot's bands between the two orange lines of sodium. The retarding
plate was 31 mm. thick and the spectroscope was a 6 m. radius Rowland
concave grating. On the original plate there are six exposures. In the
exposure of which this is a part there are over 6,000 Talbot's bands. Fig.
9 is in the same region by the same means as Fig. 3 with less of the crater
of the carbon arc focused on the slit. Both show the reversal of the
sodium lines and there are three or four bands on each. Fig. 10 was
taken with the echelon grating of 30 elements. The Talbot's bands were
produced by a 30-mm. plane parallel plate. This print exhibits both
the yellow and the green mercury lines with the bands making an angle
with the spectrum lines, for the plate was not perpendicular to the beam
of light.

Figs. II, 12, 13, 14 and 15 show prints from exposures made with the
echelon 46 mm. plate, nos. 11, 13, 14 and 15, the fringes make various

324 THOMAS E. DOUBT. [&SSS

angles with the slit, while no. 13 the bands are parallel with the slit due
to the 46-nim. plate being exactly perpendicular to the beam. No. 4
is of the indigo mercury line. The negative shows spectra of eleven orders
with Talbot's bands in each. Usually the retarding plate is not of the
correct thickness to give perfect interference fringes. If the plate is
less than the best thickness whatever be the optical arrangement for
analyzing the bands their appearance may be improved by cutting off a
portion of the beam so as to make the two portions symmetrical and
approximately equal in intensity. The Rowland concave grating with
which Figs. 3 and 4 were taken has a resolving power of about 85,000
lines in first order. When a thickness of 77 mm. was used for producing
the bands about 78 bands were visible between the centers of the two
sodium lines. If a less thickness is used the fringes would be visible and
could be improved by cutting down the two beams. To test this a large
slit of heavy cardboard was constructed with wooden cross pieces which
allowed a motion like parallel rulers. Its greatest width of opening was
15 cm. Placed between the grating and the ocular the best width for
50 mm. of glass was 10 cm., for 30 mm. about 4.4 cm., and for 10 cm.
plate 1.7 cm. of opening. Thus the definition of the fringes was improved
though the intensity of the illumination of the field was decreased.

With the concave grating the plates must be placed accurately perpen-
dicular to the line joining the slit and the center of the grating. Two
degrees out of the normal will make blurred fringes and a further slight
change will cause them to disappear. On consideration of the way in
which the interference is produced it is easy to see that inclining the plate
displaces the retarded beam so that the rays which are recombined on
the photographic plate are no longer congruent; hence there could be no
regular interference.

With the echelon, on the contrary, as the plate is inclined from the
perpendicular, the fringes are rotated with reference to the slit and
appear as sharp or perhaps even more distinct at large angles than when
parallel to the slit* The rotation may be increased until as many as
thirty bands appear on the green mercury lines and a like number on
each of the yellow and blue lines. Photographs were taken for both
positions of single order and double order. The fringes seemed to be
sharper for the position of double qrder or position of equal intensity.
When making large angles with the slit the fringes have a close resem-
blance to the photographs taken by Nagaoka and Takamine by crossing
Lummer-Gehrcke plate with an echelon. In some cases I have used as
much as 92 mm. of glass for retarding one of the beams and have observed
very sharp and clear bands. So that it is no longer appropriate to speak

Physical Rsvlew, Vol. IX.,

To face page 124

THOMAS E. DOUBT.

sS^^'] TALBOT'S BANDS, 325

of the glass used for this purpose as a thin plate. In all cases where the
thickness of the glass was 3 mm. or more I have used plane parallel glass
plates. The fact that one millimeter of glass produces one Talbot band
for about six Angstrdm units in the region of the D line suggests that we
have an objective standard of resolving power for spectroscopes. The
glass used was soft flint index about 1.6002.

There are many peculiar phenomena that can be noticed when one
examines sunlight or an arc light by means of a powerful grating with
the aid of Talbot's bands. This is especially true in the region about the
sodium lines. Their reversal and distortion may be readily observed
followed and measured by the position of the fringes which serve as a
natural fixed micrometer, provided, of course, that the plate is not changed
in temperature or position.

Theory of Talbot's Bands.

The discoverer of these bands, H. Fox Talbot, explained their formation
on the undulatory theory as an interference of the unretarded beam
with the beam retarded by the interposed plate. This explanation when
followed out will give the correct number of bands for any part of the
spectrum for a given thickness of plate.^ Let / be the thickness, /ii
its refractive index for wave-length Xi and, /i2 for X2, Xi > X2 so that
Ms > Ml- The wave-lengths in the plate will be Xi/mi and Xj/ms and the
retardations will be

and the number of dark bands between Xi and X2 will be n where

^ = 'l-x; xt)-

In photographing the Fraunhofer lines Di and Di with the Rowland con-
cave grating forming the Talbot's bands with a thick plate m is nearly
equal to m2 and if the thickness is known the mean refractive index can be
determined with considerable accuracy with a single exposure. Dr.
Wolcott Gibbs has shown that if this number is divided by the density a
value is obtained which is constant for any given substance and is mde-
pendent of the temperature.^® Hence his interferential constant is

r — - — L /M2 -- I _ Ml — I \
p p \ X2 Xi / '

where p is the density.

The simplest explanation of the lack of symmetry of Talbot's bands

326 THOMAS E. DOUBT. [^SSS

has been offered by Schuster on the basis of the pulse theory of radiation.
Since white light may be used to form the bands a single luminous impulse
should be sufficient to produce them." The action of a spectroscope
upon such an impulse assumes its most elementary form when the
analyzer is a plane reflection grating.** Let a plane light pulse fall nor-
mally upon a plane reflection grating. This grating may be considered
to be made up of total reflection strips separated by strips which reflect
no light. The light falls upon all reflection strips simultaneously. By
Huygens' principle each point in the plane of the grating may be regarded
as a secondary source. When the plane pulse falls on the grating the
secondary pulses spread out from each reflecting strip and may be brought
to a focus by a lens. The disturbance at the focus consists of a series of
pulses following each other at intervals. If the axis of the lens makes
an angle with the normal to the grating the pulses from one portion of
the grating will arrive earlier at the focus than those from the other end
of the grating. If interference is to be produced by the introduction of a
transparent plate to retard half of the light from the grating it is evident
that it mu^t be introduced on that side so as to retard the pulses which
arrive earlier. If introduced on the other side it retards those which
already arrive too late to interfere.

The plate of the best thickness would be secured if the whole series
of impulses is divided into two portions and the proper thickness of plate
introduced so as to make the pulses arrive in pairs simultaneously.
If n is the number of lines in the grating and X the wave-length for the
particular maximum corresponding to X = € sin ^ where 6 is the angle
which the axis of the lens makes with the normal, € is the grating space
then the best thickness would be iN\. The resolving power of the spec-
troscope is defined as

Hi — iti oX

for a grating where «i and «2 are the frequencies of the two spectrum
lines which are just separated, tn the order of the spectrum, and N the
total number of lines in the grating. Thus we see that the best thickness
of the plate for a grating spectroscope is equal to ^NX or one half of the
wave-length by resolving power.

It has been shown by Lord Rayleigh that the resolving power of a
prism is represented by the product of the effective thickness of the
prism by the dispersive power or t(dfjL/d\) where / is the thickness and
H is the refractive index corresponding to the wave-length X. Hence
the best thickness for a prism spectroscope would be iX/((f/n/dX). Usually
the retarding plate is not of the correct thickness to give perfect inter-

)5S!"^] TALBOTS BANDS. 327

ference fringes. If the plate is less than the best thickness the fringes
may be improved by cutting oflF a portion of the beam.

Another factor of great practical importance in resolving power is
the width of the slit, for this determines the purity of the source. In
his study of purity Schuster arrived at the conclusion that for a pure
source we must use a width of slit equal to fX/^Dt where D is the diameter
of the collimating lens and / its focal length. This width he calls the
" normal slit." Further, if this width is doubled the loss of resolving
power is only 6 per cent, but if the normal slit is eight times as wide the
purity is only 45 per cent, of the normal slit.

The theory with a half-covered circular aperture has been given by
Struve and we shall follow the earlier part of his work.^^ Let the observ-
ing telescope be adjusted for observing the Fraunhofer diffraction phe-
nomena for a point source and a circular aperture. One half of the
aperture is covered with a plane parallel plate which is transparent and
correctly placed. The plane of the opening is that of xy, the origin of
co5rdinates the middle point of the circle, the jc-axis along the refracting
edge of the plate and y-axis positive in the direction away from the
covered semi-circle.

In the focal plane of the objective let {1, 171 be the co5rdinates of the
geometric image of the origin and {, 17 the co5rdinates of the chosen
point P and the focal distance is set equal to i. Finally, let R be the
radius of the circular aperture and 6 be the phase difference between the
rays which travel through the covered and uncovered portions of the
aperture.

Then we may write the following expression for the intensity of the
point P:

/ = (Ci + Cty + (Si + 5,)«,
where

Ci = JJ cos [ Y^ - €1)^ + Y^'^ "" '''^A ^*^^'
Si = ff sin [ Y (€ " fi)^ + Y ^"^ " '''^A ^"^ ^^'

^^ " // ^'" Vt ^^ " ^'^"^ "^ Y ^"^ " "^'^A ^^ ^^'

where Ci and Si are integrated over all elements of the uncovered half
of the aperture and C% and ^2 are for the covered half of the aperture.

iWtllHi

328 THOMAS E. DOUBT.

Making the transformation to polar codrdinates with

R
3c = — r cos w,

2t _, .

— iC({ — {1) = £ COS W,

R .

y = — r sin w,
z

— R{7i — 171) =2 sin w'
Introducing these values and reducing

I COS [r cos («— «i)]d«= I cos (r cos «)d«=2 I cos (r cos «)dw,
I sin [r cos («— «i)]d«= I sin (r cos «)d«=2 I sin (r cos «)(f«,

••1

we obtain

and for

and

Ci = 2 ~ I r dr I cos (r cos «)d«,

ip f r-»

5i = 2 — I r dr I sin (r cos a))(f o),

Ci = Ci cos 5 — 5i sin 5,
5i = — Ci sin 5 *— 5i cos 5,

Ci cos 5i sin - I .

It is remarkable what this expression exhibits. Convenient as a
complete square it is valuable not only for a circular aperture but also
for other apertures so long as the limiting line of the plate divides the
opening into two symmetrical portions. In case the semicircles are not
always symmetrical but are separated by a small strip of width e, then
a correction must be made to the above, and in the place of 6 write
d + e{z/R) sin wi wherever this variable appears in the formula.

As a special case if the geometrical image of the point P lies in the axis
rju then f = f 1 = o, and «i = ir/2, the intensity at the point Pi will be

2 C^

Jo{z) = — I cos (2 cos «)(/«,
tf Jo

JJ^^*] TALBOT S BANDS, 329

and

zJi{z) = I zJo{z)dz,

»»/2

2 r"^

Ho(z) = — I sin (z cos «)(fa>,
zH\{z) = I zHo{z)dz,

/ =

1 7i(2) cos - - Hi{z) sin - 1 ».

where

^ = Tn (^ "■ ^i)«

Ji(z) is the Bessel function of order unity which is well known in both of
the following forms

2 f * . 2 f'^*

•'^iC^) = I sin (z sin w) sin cjdu) = — I cos (z sin «) cos* wd«.

X»/o ITS Jo

In a similar manner the unsymmetrical H functions may be derived.
Through the double differentiation of Ho{t) it yields

d^Hojz) . ,dHo(z) . ,,,, 2
but

and from this according to definition

Hi(z) = — 1 1 — I cos (z cos «) cos wdo) I

4 r^. ,/gcosa)\ 4 r"" . . /gsina)\ .

= — I sm* I I cos« aw = — I sm* I I sm wa«.

ttJq ^\ 2 / li k/'rJo \ 2 /

A second definite integral may be derived from this by use of Fourier
series:

f{z) = "" I sin {az)da I /(X) sin (aiX)dX
where

/(.) - ^

(sin a)\
~ Ij I sm zaaa 1 sm waa> 1

Z IT ^Q Jn Jn \

330 THOMAS E, DOUBT,

but since

. , /Xsin wV ^

m Sin a\ sin' I I I ^

Hxiz)

\ 2 / _ —when sin w > a,
^ dX = ^4

[ o when sin « < a,
= — I sin azda I sin wdo) = "" I sin az^l — a^(fa,

'T 4/0 */gln-J « ^4/0

and from this finally

Hi(z) = -~ I sin (s sin «) cos* wdco.
^ Jo

Incidentally it may be mentioned that this furnishes the means for com-
paring the first form of Ji{z) with the second and this method also allows
us to exhibit the fundamental property of Bessel's functions as a very
simple derivation.

^ MZ) = 7n-lW + A+l(2),

z
dJnjz) T r \ T r \

2 ^^ = Jn-l{z) - //lilW.

Next substitute for a single point source monochromatic light a series of
independent sources whose geometrical image lies on the 17 axis. The
points shall only be distinguished by their wave-lengths which from
one side to the other may be taken to determine the specific intensity.
On such a basis there would exist at each point an ideal vanishingly
small spectrum at right angles to the division of the two portions of the
objective. The intensity /(17) of the points in the geometric image of
this spectrum may be found through the summation of the intensity /
for all possible values of 171 which lie between the limits — 00 and + «
so we may take for this

Jdrju

m = I

or through substitution of

2tR

z = -T— (^ — »?i).

I(v) = 2t}JP J ^-^ COS- ~^^ 2/

Wherein the phase difference 6 is a function of the wave-length and hence
also a function 171 or z which one may evaluate from Taylor's theorem
in powers of 2. If one designates with 60 the phase difference in the

IfV^^] TALBOTS BANDS, 33 1

point 17 we arrive at an approximation through

« - fi. + (ni - n) (4;)" = «• - 5^(1^)^

= 5o + az.

In which the higher powers of z may be neglected. Ji(z)/z and Hi(z)/z
are by nature very small quantities. Since this contribution to the value
of d would be very small so on the same ground we will assume that the
specific intensity of the spectrum will be constant for each X (and in the
same expression for a) in the point 17 for which the wave-length is taken.
Omitting the factors 2tXjR' the intensity of the point 17 for which the
phase difference is 60

From this the law of the intensity of the maxima and minima remain to
be determined. Differentiating this expression for 60 and putting the
differential coefficient equal to zero, one perceives that a varies slowly
with 17 for do a constant; hence sin do = o, from which it follows that the
maximum and minimum respectively exist only where the path difference
for the light rays are an odd or an even number of half waves.

The results obtained from the consideration of these integrals show
that for a circular aperture it is impossible to obtain absolute maxima
and minima otherwise the conclusions of Sturve are the same as those
Lord Rayliegh has given in his article on Wave Theory in the Encyclo-
paedia Britannica. The explanation of the rotation of the fringes in
using the echelon may be carried out according to Wood's suggestion
that the plate used for forming Talbot's bands may be considered as an
echelon of two elements.*^ The expression for the intensity as found by
F. B. Galitzin is

^ M ti / \nsm (£/2)/ '
or

I ^ IoA(u)ip{v).

Where n is the number of steps in the echelon, s width, u = Tqs/X^

q = sin (<p ^ 6) + sin ^, where 6 is the angle of incidence and <p the

angle of refraction

a a

£ = /) + 2t ^ 5 + 2t f + tan ^,

* A A

/> = 27r I - j cos (^ — ^) — /i sec ^ + tan ip sin (9 — B),

332 THOMAS E. DOUBT, [^»

The expression for / above indicates that it is always positive, periodic
and that it varies slowly with the angles. As one changes the angle of
incidence B the spectra shift in such a manner that there is one position
for which the corresponding angle ^ is a minimum.

Summary.

The resolving power of a spectroscope may be represented by the thick-
ness of the plate of glass used for forming Talbot's bands. The best
thickness represents the highest power for the given instrument. Ex-
amples are given here for resolving powers represented by less than one
millimeter up to 92 millimeters in thickness. This furnishes further ex-
perimental evidence for the inference that for infinite resolving power
infinite thickness would be required.

Stokes speaks of retardation of several hundred wave-lengths and says
that: '* This exalts our ideas of the regularity which must be attributed
to the undulations."* My photographs show retardations of many
thousand wave-lengths. Since the work of Lord Rayleigh, Guoy, Michel-
son, Fabry and Perot, and others little need be said about " regularity "
in radiation. However, the phenomenon of Talbot's bands raises the
question whether white light is not as much a unity as monochromatic
light and withal just as simple? The retarding plate enables us to apply
one test to all wave-lengths, while the spectroscope enables us to apply
another test.

Much of the experimental work was done at the University of Chicago.
I am pleased to express here my grateful acknowledgments to Professors
Michelson, Millikan, and Gale, and the staff of Ryerson Physical Labor-
atory for the facilities so freely placed at my disposal.

Bibliography.

The first announcement of Talbot's bands appeared in a brief note in the Philosophical
Magazine for 1837.

I. Phil. Magazine. 1837, p. 364, LXXIL, 3d Series. 10. Mr. Talbot's Experiment on the
Interference of Light. By H. Fox Talbot, Esq., F.R.S. It was translated into
German and appeared in the Annalen der PhysUc und Chemie von J. C. Poggendorf
4w s.wer., 1837. Sir David Brewster tonounced before a section meeting of the British
Association in 1838 that these bands could not be formed except by introducing the
retarding plate from the violet side of the spectrum.

a. On a New Kind of Polarity in Homogeneous Light. By Sir David Brewster. Report
of the British Association. Vol. 7, p. 13, 1838.

3. On the Theoretical Explanation of an Apparently New Polarity in Light. By G. B.
Airy, Esq., M.A.F.R.S., Astronomer Royal presented to the Royal Society, June 18,
1840. This is the Bakerian Lecture for 1840.* In it Sir David Brewster is called the
"Father of Modem Experimental Optics." It shows the calculations of the intensities
in different directions as seen by an eye too near or too far for distinct vision. Values
are given both in the form of curves and in tables calculated by Fresnel's integrals in the

Na*] . TALBOTS BANDS. 333

theory of diffraction. In a supplementary note values are given for the bands seen
in focus.

4. Airy. Phil. Trans., p. i. 1840. Sir G. G. Stokes saw that these bands might be of great

value in determining refractive indices and the laws of dispersion. He gave a splendid
treatment of the theory which has been followed ever since whenever the bands have
been dealt with by analytic methods.

5. On the Theory of Certain Bands Seen in the Spectrum. George Gabriel Stokes. Phil.

Trans., 1848. p. 227. Also his Mathematical and Physical Papers, Vol. 2. p. 14. It was
his method that Lord Rayleigh has followed and abbreviated in the article. "Wave
Theory." in the Encyclopedia Britannica.

6. Wave Theory.— Lord Rayleigh. Enc. Brit. 9th Ed.. XXIV.. 1888. Also Sc. Papers.

Vol. III., p. 123; also i^-tide on Diffraction. Vol. VIII.. p. 249. Enc. Brit., iith ed.
Application of these bands was made by Eisenlohr. following a suggestion by Helmholtz.
to the determination of wave-lengths in the violet and ultraviolet portions of the
spectrum.

7. Eisenlohr, W.. Wirk d. violette u. ultra-violetten LichU, Vol. 93» P- 623. Die brech-

barsten od. unsichtbat Lichtstrahlen im Beugungsspektrum u. ibre WellenlSluge 98,
353* Wellenliluge der brechbarsten und der auf Jodsilber chemisch wirkenden Strah-
len. Pogg. Ann., 99. 159, 1856.

8. Mascart. Journal de Physique t. Application du Spectroscope a L'Observation des

phenomenes D' Interference. Fig. i, Plate i.

9. Experimentelle Prusung der Airy'schen Theorie der Talbot'schen Streifen von V. Dvorak*

Pogg. Ann.. CXLVIL, S. 604-615. 1872. Sur Theorie der Talbot'schen Streife von

V. Dvorak. CL., S. 399-410, 1873, Pogg. Ann. Following out lecture demonstrations

given by Mach.
10. Dr. Wolcott Gibbs showed how one may determine an optical constant from liquids

which is independent of temperature by merely counting the number of Talbot's

bands between two fixed lines of the spectrum. Optical Notices in Pftxreedings of

Amer. Acad. Arts and Sciences, Vol. X., 1875, p. 401.
IX. Cber die durch planparallel Krystall platten hervorgemfenen Talbot'schen Interference-

streifen. Von L. Ditscheiner. Berichte der Wein. Akad.. 1868. LVII., S. 709-734.

iiher einige neue Talbot 'sche Interfemze Erscheiningen. Von. L. Ditscheiner, Berichte

der Wien. Akad.. S. 529-553, 1871, LXIII., 1871, Ab. 2.

12. Application de Franges du Talbot a la Determination des indices Refraction des Liquids.

M. Hurion. Jour, de Physique, Vol. 10. p. 154, 1881.

13. Ueber die sceinbare Polaritilt des Lichtes bei den Talbot'schen Linien. Von B. Walter.

Annalen der Physik und Chemie. — ^Wiedemann, 39, 1890.

14. Sur la Thterie des Bande de Talbot. Par M. Carimey Joumalde. Physique 2** serie,

t. VII., p. 60, 1888. See also Thtorie des Ph^nom^neo de Diffraction observes a
L'Infini ou au Toyer D'Une. Lentille M. Joubert. Journal de Physique, i" serie.
t. III., p. 267, 1874.

15. Method pour mesurer en longuers d'onde petites epaisseurs. M. J. Mac^ de L^piney.

Ann. de ch. et de ph., 6* s.. t. X., 1887. pp. 68-85. Mesures Optiques D'Etalous
D'Epaiseur 6 es., t. X., 1887, pp. 216-255. Sur une nouvelle determination de la
masse du decimetre cube d'eau distillie privee d'air a son maximum d'intensite. Par
M. J. Mack's de L^pinay. 1897, 7* st.. XL, pp. 102-114. See also his book.

16. Franges d'interf^rence. C. Naud, Editeur, Paris, 1902. Interference methods of

extreme accuracy. Fig. 3, Plate I.

17. The only one who has worked out the complete theory of Talbot's bands for a circular

aperture is Hermann Sturve in Zur Theorie der Talbot'schen Linen, M^moires de
L'Acad^mie Imperiale des Sciences de St. Petersbourg, VII*. Serie, Tome XXXI.,
No. I, 1883.

18. Sue le calcul des franges de Talbot. Par E. Bichat. Bibliotheque Universalle Archives

des Sciences Physiques et Naturelles. Gendve 26 (1891), 5.

334 THOMAS E, DOUBT. ^ [|^g^

19. An explanation of Talbot's bands on the ether pulse theory of light is given by Arthur

Schuster in A Simple Explanation of Talbot's Bands, in Phil. Mag., Vol. VII., p. i,
1904. See also his Theory of Optics, ad ed., p. 119, 1909.

20. Walker. — Phil. Mag., 1906, 6th Series, p. 631, Vol. II.

21. R. W. Wood photographed the bands with an echelon grating. The resolving power of

the echelon he used was about 500. Phil. Mag., Series 6, Vol. 18, 1909, pp. 758-767.
For resolving power of his echelon see pp. 627-629, Phil. Mag., Vol. i, series 6, 1901.
A Mica Echelon Grating.

22. H. Nagaoka and T. Takamine. Crossed Spectra obtained by Combinations of Different

Interferometers and their Applications to the Measurement of Difference of Wave-
length. Phil. Mag., Series 6, Vol. 27, Jan.-June, 1914.

References in Books.

23. Mascart. Trait6 D'Optique, p. 473.

24. Winkelmann. — Optik, Band VI., S. 1084.

25. Wullner. — Die Lehre von der Strahling vierter B and Lehrbuch der Experimental Physik.

s. 650-655-

26. Kirchhoff. — ^Vorlesungen Uber Mathematisch Optik, s. iii. 1891.
37. Kohlrausch. — Lehrbuch Praktischen Physik, 9th s., 259.

28. Kayser. — Handbuch der Spectroscopie. I. Band, s. 737.

29. Pickering. — Physical Manipulation, Vol. 2. p. 304.

30. Wood. — Physical Optics, 2d ed.

31. Preston. — Theory of Light, 4th ed., pp. 171, 174, 276.

32. Schuster. — Theory of Optics, 2d ed.

33. J. Mac6 de L^pinay. — ^Franges d'interference.

34. R. A. Houstoun. — A Treatise on Light, p. 379.

IfS!'^'] EMISSION OF ELECTRONS. 335

THE EMISSION OF ELECTRONS BY A METAL WHEN
BOMBARDED BY POSITIVE IONS IN A VACUUM.

By W. L. Chenby.

IN the following paper, experiments are described which were under-
taken with the object of ascertaining how the number of negative
electrons emitted by a metal when bombarded by positive ions in a
vacuum depends on the number, the velocity, and the nature of the
positive ions.

It is well known that when a metal is bombarded by positive ions of
sufficiently high velocity' it emits negative electrons.^ One of the first
to show this was Villard,* who found that cathode rays are formed by
positive ions impinging upon the cathode. He placed near the cathode a
diaphragm having two small holes. As the tube was gradually ex-
hausted, so long as the dark space did not extend to the diaphragm, the
current flowed uniformly from the whole surface of the cathode. But
after the dark space extended beyond the diaphragm the emission from
the diaphragm became concentrated at two points opposite the holes in
the diaphragm. In a high vacuum, two narrow rays passed from the
cathode through the holes and produced a shadow of the diaphragm on
the walls of the tube near the anode, showing that electrons were formed
only when the positive ions hit the cathode.

J. J. Thomson' was the first to observe that when alpha rays from polo-
nium bombard a metal, many slow speed electrons are emitted. Thom-
son named these negative electrons " delta rays," and concluded that
their velocity was about that acquired in falling though a few volts only.

Ftichtbauer has^ shown that negative rays are given off when a metal
is hit by canal rays, and that the velocity of the negative rays is inde-
pendent of the velocities of the canal rays. He has further shown that
some metals also reflect canal rays. When the velocity of the canal
rays are due to a P. D. of 30,000 volts he found that all metals give off
electrons for each canal ray particle in the same order as Volta's series;
platinum giving least, and aluminum four electrons for each canal-ray
particle.

» Townsend, Electricity in Gases (igis).

* Villard, Journal de Physique (3). 8, p. i (1899).

» J. J. Thomson. Proc. Cam. Phil. Soc, 13. p. 49 (1904).

*C. FUchtbauer, Phys. Zeit., Vol. 7. PP- i53-i57 and pp. 748-750 (1906).

336 W. L. CHENEY, [g^^

Campbell found the speed of the delta rays to be independent of the
speed of the alpha rays by which they were excited, and independent of
the material from which the rays are emitted.

Bumstead* has found evidence that in addition to delta rays, positive
ions are also produced when alpha rays impinge upon a metal in a very
high vacuum. These, however, appear to come from the layer of
absorbed gas on the metal.

More recently, McLennan and Found* have investigated the problem
by measuring the number of delta rays emitted from zinc when bom-
barded with alpha rays in a high vacuum. They found an emission of
three electrons per alpha particle from freshly scraped zinc. This effect
diminished with the lapse of time and ceased altogether for a while when
the zinc was freshly coated in vacuo with a deposit from zinc vapor.

The Experiments.

The method employed in this investigation was to obtain positive ions
by heating different salts, such as potassium sulphate, in a vacuum, on a
strip of platinum, through which an alternating current was passed. A
metal plate was placed near the strip and the positive ions made to bom-
bard it by giving it a negative charge. The current between the strip
and plate could be easily measured by a sensitive galvanometer.

The ratio of the negative electrons emitted by the plate to the positive
ions striking it, could be found in the following manner. Let Ci represent
the thermionic current carried by positive ions and negative electrons,
so that
(i) Ci = C+ + C-.

Now, if a transverse magnetic field be set up which will stop the electrons

by causing them to curve back upon the metal plate but will not stop

the positive ions,

(2) C, = C+.

Dividing (i) by (2)

Ci ^ _ a.
c% c+

Let iV+ be the number of positive ions striking the plate per second, and
iV_ be the number of negative electrons given off from the plate per

* Campbell, Phil. Mag., Vol. 22, p. 276 (191 1), and Vol. 23, p. 46 (19x2).

* Bumstead, Am. Journ. of Sci., Vol. 36, pp. 91-108 (1913).
' McLennan and Found, Phil. Mag., Vol. 30, p. 491 (1915).

Vol. X.1
No. 4. J

EMISSION OP ELECTRONS,

337

second. Then C+ = N^^ and C- = N^^ where «+ and e^ are the
charges on the positive and negative ions respectively. But since e^. = tf-,
we have the desired relation,

(4)

The procedure was to observe, for a given P.D., first Ci (directly with
the galvanometer), then C2, then C2 with the magnetic field reversed, and
finally Cu again. This was done to obviate any fluctuations arising from
a change of heating current. In most cases, however, this was really un-
necessary, for the initial and final values of Ci did not differ appreciably.
From the means of Ci and C2, C^/C^ was calculated. Representative
values to illustrate this are incorporated in Table I.

Table L

Pt Cathode, K + ions.

P. D.

Voltt.

Oalv. Deflections in Mm. (1.65 X io~*® Amp.).

Ci/G.

Ci (N Re-
versed.)

Mean.

C-IC^

146
190
250
280
350
400
475
525
560

131
147
154
160
166
166
172
168
152

130
146
152
157
163
162
168
164
148

130
146
156
157
163
162
168
164
148

130
147
158
159
166
165
172
169
152

130.5

147

156

159.5

166

165.5

172

168.5

152

130
146
154
157
163
162
168
164
148

1.004
1.006
1.012
1.015
1.018
1.021
1.023
1.025
1.025

.004
.006
.012
.015
.018
.021
.023
.025
.025

The magnitude of H (the magnetic field), necessary to stop the negative
ions without stopping the positive, could be calculated from a formula
given by J. J. Thomson^ for determining e/m when using a magnetic field
to stop ions passing between parallel plates; viz.,

e
m

where e/m is the ratio of the charge in the ion to its mass, V the potential
difference between the plates, H the magnetic field, and d the distance
between the plates.

Calculations according to this formula are exhibited in the following
table.

1 J. J. Thomson, Conduction of Electricity through Gases (1906), p. 219.

338

w. l. cheney.
Table II.

NBtoT* or ton*.

RlKtroDt.

K t^lcni.

U + Ions.

n + ion*.

,1m.

Kii"

iLli%.

eS^.

i^^.

H(calculated) neceaeary to just stop ions

volts (10" E.M.U.)

H(calcuUted) necessary to just stop ions
volts (6X10" E.M.U.)

at 100
at 600

50
110

9,000

21,000
400

4,000

9,000
400

13.000
32,000

The effect produced by H was further tested by choosing a definite

P.D. and varying H over a considerable range (ioo-i,ooo). As no

change occurred in the diminution of the leak it was concluded that

even with as high a value of /f as l ,ooo units positive ions were not being

deflected unless it were at very low potentials, such as ro-50 volts. No

observations of CJC+ were made at these low P,D.s, the reason being

that any diminution of current

caused by H was too small to be

detected or did not exist at all.

In nearly all the observations

here recorded H was 400 units.

The apparatus is shown dia-
grammatically in Fig. r. It con-
sisted of a glass tube about 4
cm. in diameter cemented to a
brass plate P with sealing wax,
and supported between the poles
; L. of a lai^e electromagnet. SS
J were two brass rods, supporting
J the narrow strip of platinum, A.
One of the rods passed through
an ebonite plug and was thus in-
sulated from P. The strip of
platinum could be heated to any
desired temperature by passing
a 6o-cycle alternating current
through it. It could be made

Fig. 1.

the anode by connecting to the positive terminal of a battery of
" Tungsten Ever-Ready " cells (capable of giving nearly 600 volts),
while C, the metal plate, was made the cathode. C was carried by a
micrometer screw and could be moved up and down by turning the ground
joint J, so that the distance AC could be varied as desired. Some ob-

Vol. X.
No. 4.

]

EMISSION OF ELECTRONS.

339

servations made by moving Cup and down and noting the current showed
that the leak across the gap AC decreased slightly as the distance AC
was increased. This is shown in the following table.

Table 1

[II.

Diat. Between

Blectrodes

(Mm.).

Current X (1.65 X lo-w Amperes) (P.D. ^ 146 Volts).

160
160
154
150
150
145
142

160
158
156
155
154
148
145

150

140
138

134

134
132

127
127

126

127
125

120
118

112

180
178

170
168
166
165

210

208

202

200

198

196

In the subsequent observations the distance between the platinum strip
and the metal plate was kept at about one centimeter.

The potential difference between A and C was measured by a Kelvin
Electrostatic Voltmeter, and the thermionic current was measured by a
Leeds and Northrup sensitive galvanometer (sensibility = 1.65 X lO"^**
ampere per mm. deflection).

The greatest difficulty throughout the experiment was the securing of
a good vacuum. A Gaede rotary mercury pump was used in series with a
box pump. When a sensitive McCleod Gauge indicated no gas pressure,
the vacuum was put to further test by means of an induction coil whose
terminals were placed across the gap AC and the pump kept running

until no fluorescence appeared in the tube and the spark preferred to
pass through the air outside. The mercury or oil vapors which might

340

W, L. CHENEY.

tSlCOMD
Sbubs*

have been in the apparatus were frozen out by means of COi snow.
Finally, the pump was kept running throughout a series of observations.
The apparatus was kept dry by means of P2O6, and the vacuum was
washed out from time to time with a little dry air.

To test the variation of the thermionic current with the change of
potential difference, a double throw switch was placed in the circuit so
that the leak for any P.D. could be compared quickly with the leak for
15 volts. This is illustrated in Fig. 2, in which the P.D.s are plotted
as abscissae and the ratios of the leak for given P.D.'s to the leak for 15
volts as ordinates. It is worthy of note that with low potential differ-
ences the leak rose rapidly with the increase of P.D. until about 150 volts
where it approached saturation.

To find CJC^, observations were made with aluminium and platinum
as cathodes and K2SO4, LiaSOi, and RbjSOi respectively on the hot
platinum strip ^4, as a source of positive ions. For each particular salt
and metal, a great many observations were made. Table IV. shows
representative values for a number of observations in a particular case,

Table IV.

Pt Cathode, K + Ions.

P.D.

(Volt.).

C^IC^

Mean

146

.005

.007

.004

.002

.005

.003

.005

.003

.007

.0045

190

.005

.012

.006

.008

.005

.007

.009

.006

.009

.004

.007

250

.012

.013

.012

.020

.012

.011

.0125

280

.012

.015

.012

.015

.018

.016

.015

.013

.012

.013

.014

350

-.018

.032

.018

.013

.012

.020

.012

.016

.019

.013

.016

.017

400

.023

.014

.023

.021

.025

.018

.025

.015

.013

.021

.016

.020

.0195

475

.023

.020

.022

.018

.030

.024

.015

.022

.021

.017

.020

.016

.021

525

.020

.030

.025

.026

.022 .022

.025

.028

.018

.028

.018

.024

560

.020

.026

.032

.025

.028

.028 .028

.031

.018

.028

.029

.025

while Table V. gives the mean values for all the salts and metals used.
By an inspection of Table V. and Fig. 3, it is seen that the largest effect
occurs in the case of lithium and the smallest with rubidium, while that
from potassium lies between the others; the values for all three being
greater with aluminium than with platinum.

Some preliminary experiments showed that the effect decreased some-
what after the apparatus had stood evacuated for several days. Con-
sequently, all the results recorded in Tables I., IV. and V. are those taken
after the apparatus had been allowed to stand evacuated for several days.
Some observations were made with Li2S04 on the hot strip, and alumin-
ium as cathode, after the apparatus had been standing for nearly a

Vol. X.!
Na4. J

EMISSION OF ELECTRONS.

341

Table V.

Mean Values of G/C+,

Al Cftthode.

Pt Cftthode.

P.D. Volts.

AV.

Lt\.

Z/+.

Rh^.

,004

1 • • •

• • •

• • • •

.006

• • • •

115

» • • •

1 • • •

> • • •

• • • •

.0016

125

.008

> • • •

1 • • •

• ■ • •

• • ■ •

• • • •

146

.011

1 • • •

» • • •

.0045

.013

.002

177

1 • • « I

.010

1 • • •

• • • •

190

r • • •

1 ■ • •

1 • • •

.007

.017

.003

215

.018

.020

1 • • •

• • ■ •

• • • •

• • • ■

235

> • • ■

» • • •

1 • • •

• • • •

.005

250

t • • •

» • • •

> • • ■

.0125

• • • •

• ■ • •

265

1 • • •

• • •

• • • •

.019

• • • •

275

.025

.058

.008

.014

.007

• • ■ •

300

> • • •

1 a • •

1 • • •

.024

• • • •

• .,. •

340

.037

.093

.014

• • • •

350

> • • •

1 • • •

> • • •

.017

• • • •

.008

360

r • • •

t • • • 4

1 • • •

• • • •

.026

• • • •

375

.047

» • • •

• • • •

390

> • • •

> • • • <

• • •

• • • •

.030

■ • • •

400

> • • •

» • • • 1

1 • ■ •

.0195

• • • •

.009

450

.067

.138

.022

• • • •

475

.080

> • • • 1

k • • •

.0207

.036

• • • •

535

1 • • • t

.214

.036

• • • •

• • • ■

• • • •

560

.092

> • • • <

• • •

.025

.043

.012

T%. f%

.20

i •

fli^K

tiia 1

. /

fiS

—

.05

/ /

— —

['^

JSJ--'

<2.

£.

a__

^^*

— "V^

200

400 eoo

PD. in ¥olU'

200

400

600

Fig. 3.

342

W. L. CHENEY,

rSBCDMD

LSbexbs.

month. Comparing the results with those obtained after a week's
time a slight decrease is noticeable (Fig. 4).

It seems quite likely, therefore, that part of the effect, at least, is due
to gas absorbed by the metal. However, some tests were made with

^^^^

,mf —

—J

f «

-/.

J J —

—^

1
^

—

,10 —

/•

JC -

•

)

\s.

100

200

300

400

500

Fig. 4.

platinum cathode and rubidium, after heating the metal plate for several
hours in air at a low pressure by bringing it in contact with the hot plati-
num strip. The results obtained (Table VI.), although somewhat erratic.

Table VL

Showing the Effect of Gas, {Pt Cathode, Rb^ Ions,)

t-/C

P.O. (Volts).

Aa in Table V.

lis

146

1<)0
235
265
275
350
300

-kX)

4S5

5Kl

.0016
.002
.003
.005

• • • •

.007
.008

• * • «

.009
.011
.012

After Heatinc in Air.

Alter Heatinir in
Hydrocen.

.000

.0025

.004

.005

.0065

.0085

.007

• • • «

* « • •

.009

.O-JS

.010

.Oi)Q

« • « «

.OOQ

.012

.OvWS

.014

.010

.017

^^^'] EMISSION OF ELECTRONS. 343

approximate to those obtained prior to the test with air. A similar test
was made using hydrogen instead of air. This time the values of C_/C+
were slightl}' greater than previously, indicating that the platinum had
possibly soaked up some of the hydrogen. These values, however, are
not as greatly in excess as one might expect, so it appears that the gas
was only slightly absorbed.

Application to Discharge in Gases.

Consider the dark space in a discharge of electricity through a gas at
low pressure and suppose no positive ions striking the cathode during a
particular time interval. Some of these positive ions striking the cathode
will set free electrons but only those which have fallen through a long
enough free path to acquire sufficient velocity. Let the mean free path
be represented by X. Let n be the number during this time interval
which have free paths greater than a length x. Then dn = — findx,
where P represents the number of collisions a positive ion makes in going
one centimeter.^ Therefore, on integrating, n = noc~^* = «oc~*^^. The
number having free paths between x and x + dx is therefore given by

dn^'^r'i^dx.
X

Let 7 be the ratio of the number of electrons emitted from the cathode
to the number of positive ions producing them. 7 is a function of the
velocity of the positive ions as has been found in the experiments de-
scribed above. 7, the average value of 7, is given by the expression

By considering Fig. 3, it can be seen that the curve representing the
values of 7( = C^/C+) is practically a straight line and can be expressed
analytically by

(6) y = aV -b,

where V is the potential difference, a the slope of the curve and b a con-
stant. Equation (6), however, holds only for positive values of 7.

If we assume, on the basis of Aston's experiments,* that there is no
appreciable difference of potential between the cathode and the adjacent
gas, we may write

V -- r Xdx,

> Townsend, loc. dt.

*F. W. Aaton, Proc. Roy. Soc, i4. 84, p. 526, 1911.

344

W, L. CHENEY,

fSSCOND

LSbribs.

where X is the electric force. In the dark space X varies uniformly,
being a maximum at the cathode and a minimum at the negative glow,
so that X — A — Bx, where A and B are constants. At the negative
glow X = o, and A = JBD, where D is the length of the dark space.
Hence

(7)

V = B r {D - x)dx = b{^Dx - j).

When -i = o, V = b/a. Call this particular value of V, V. Then

'" Bx"

{D - x)dx = BDx' -

(8)

' = bJ (D - x)dx = BDx' -

where x' is the distance in the dark space representing the paths through
which the positive ions fall under the P.D. of V volts before impinging
on the cathode.

To determine B, take

= I xdxt
Jo

where K is the " normal " cathode fall of potential. From this we get

.H0t

Ai n Jf

-HOLECilLAfL WB/GHT "
Fig. 5.

B = 2K/D^ and equations (7) and (8) now become

(7')

F =

(- - ?) .

(8') '^ ' h ("'' - t) ■

Equation (6) becomes

(6')

2A/„ X^\

y = a^[Dx---)-b.

KL.X.1

>. 4* J

VOL.X.1

No.

EMISSION OF ELECTRONS.

345

and finally (5) becomes

Suppose now we consider a special case, viz., an aluminium cathode and
a discharge through hydrogen with a gas pressure corresponding to one
mm. of Hg. Investigators have found that under these conditions,
D = 1 cm. approximately, K = 200 volts (nearly), and X (according to
Meyer) is .013 cm. for the hydrogen molecule. The values of F', a,
and 6, corresponding to the different kinds of positive ions, are found from
Fig- 3i when aluminium was used as cathode.

Positive lont.

Mol. Wt.

7
39

.0006
.0002
.0001

.09

.015

.020

150

200

Similar values for hydrogen are found by plotting the above values of
a and b against the molecular weights (Fig. 5) and extrapolating, a is
found to be .007, b = .104, and F' = 150 volts (nearly). Substituting
the values of K, V, and D, in (8') and solving, x' = .5 cm. Substituting
for the various constants their numerical values, (5') may be simplified to

Integrating
(9) 7 = €"'^1- .28(x + X) + .I4{^ + 2X(jc + X)} + .I04l!5.

Upon evaluating, 7 is found to be of the order io~^^, which, of course, is
negligible.

While Aston has found no appreciable difference of potential between
the cathode and the gas, others^ have found a considerable drop in the
potential right at the cathode. Under these conditions.

-r

Xdx + F„

where Fo denotes the fall of potential right at the cathode and

•/O

Xdx + iCo,

where Kq denotes the value of Fo in the case of the ** normal " cathode
fall.

» C. A. Skinner, Phys. Rev., June and Aug., 1915, W. L. Cheney, Phys. Rev., Feb.. 1916;
W. E. Neuswanger, Phys. Rev., Feb., 1916.

346 W. L. CHENEY. [^^

Thus equations (7'), (8'). (6')i and (sO become modified to
(10) V = ^-pi '- [Dx --)+ Vo,

x' is found to be .3 cm., Ko (from the experiments of the writer) is
neariy 90 volts; the values of the other constants are the same as above.
7 is in this case of the order io~^®.

Thus, it appears from calculations based on either Aston's or Skinner's
experiments, that in the case of the " normal " cathode fall of potential
in a discharge of electricity through hydrogen at low pressure extremely
few electrons are set free from the cathode. The above calculation,
however, is subject to error since one is not quite sure of the value V.
As already stated above, it was difficult to obtain any accurate observa-
tions of 7 for low values of V and one is not certain that the curves in
Fig- 3» which we have assumed to be nearly straight lines, do not become
asymptotic to the F-axis. At any rate, the calculation shows that 7
is very small.

Skinner,^ working with the ** normal " cathode fall in hydrogen at
low pressures, calculated 7 to be of the order lo"*, for an aluminium
cathode. The writer,^ making use of Skinner's theory, found under
similar conditions the same order of magnitude for 7.

Skinner's theory, however, does not take into account the collisions of
positive ions with the molecules of the gas. H. A. Wilson,' taking into
the account the ionization by collision of the positive ions has shown that
7 is probably small.

Townsend consider 7=0 except at very low pressures when high
potentials are necessary. Aston* has made some investigations under
these conditions. Here is one set of values which he obtained when using
aluminium cathode in hydrogen: V = 700 volts, D = 2.09 cm., p = .157
mm. of Hg. From these values x' works out to be .25 cm. and X = .083
cm. Applying these to equation (5') 7 is found to be of the order io~*.

> C. A. Skinner, loc. cit.

* W. L. Cheney, loc. cit.

» H. A. Wilson, Phys. Rev., Sept., 19 16.

<F. W. Aston, Proc. Roy. Soc., A, Vol. 87, p. 437.

li^A^'] EMISSION OF ELECTRONS. 347

It appears, therefore, that y ing-eases very rapidly as the pressure is

diminished. The experiments described in this paper show that in the

case of a thermionic current in a vacuum y is appreciable for lower

values of the P.D. corresponding to the " normal *' cathode fall in

hydrogen.

Summary.

1 . The magnitude of the thermionic current in a vacuum corresponding
to various P.D.s has been compared with the thermionic current cocre-
sponding to a P.D. of 15 volts.

2. The ratio of the number of electrons leaving the cathode to the
number of positive ions striking it has been found with positive ions of
different velocities and for two different metals, viz., aluminium and
platinum. It has been found that this ratio depends on the velocity of
the positive ions.

3. It has been found that the effect is diminished somewhat after the
metal has stood in a vatcuum for some time and increased slightly after
it had stood in hydrogen. It appears, then, that the effect is at least
partially due to gas in the metal.

4. The ratio of the number of electrons emitted from the cathode to
the number of positive ions bombarding it has been calculated for the
case of a discharge in hydrogen.

The writer wishes to express his indebtedness to Professor H. A.
Wilson, at whose .suggestion this investigation has been undertaken,
and whose interest and kindly counsel have been very valuable in sur-
mounting many difficulties.

The Rice Institute,
Houston, Texas,
March, 1917.

348 H. L. BOWES AND D. T. WILBER. [^

THE FLUORESCENCE OF FOUR DOUBLE NITRATES.

By H. L. Howes and D. T. Wilbbr.

•

IN an early paper by Professors Nichols and Merritt^ on the lumines-
cence of the uranyl salts it is noted that the effect of the water of
crystallization on the spectrum of the uranyl nitrate is to shift the bands
slightly in the direction of the longer wave-lengths. In a more recent
paper* the effect on the fluorescence of the nitrate has been studied in
detail and it is shown that the spectra of the anhydrous salts and of
specimens with 2H2O, 3H1O and 6HjO differ from each other profoundly
as regards the position and the grouping of the bands.

On the other hand there is good reason to think that crystal form has
an important bearing upon the structure and arrangement of fluorescent
spectra. In a study of the polarized spectra of four of the double uranyl
chlorides,' which crystallize in the triclinic system it was found that the
spectra of these salts were almost identical in arrangement and in the
absolute position, relative intensity and resolution of their bands. In
an independent investigation of frozen solutions of various uranyl salts
two or more strikingly different spectra from uranyl nitrate were obtained
by varying the rate of freezing the aqueous solution.* A year ago the
present authors made a brief study of seven forms of the sodium uranyl
phosphate and concluded that only when crystals were found could a
resolved spectrum be obtained by cooling.*

The object of this brief paper is to throw a little more light on the r61e
played by crystal structure. We had hoped to be able to produce two
crystals having different crystal systems but identical chemical formulae.
In this we have failed, but have two crystals with nearly the same formula
and different crystal systems.

The two pairs of double nitrates studied are mono-ammonium uranyl
nitrate NH4UO2 (N05)8; di-ammonium uranyl nitrate (NH4)jU02(NOi)4
2H2O; the mono-potassium uranyl nitrate KUO2 (NOg)i and the di-
potassium uranyl nitrate K2U02(N03)4.

» Nichols and Merritt, Phys. Rev. (i), XXXIII., p. 375, 191 1.
» Nichols and Merritt, Phys. Rev. (2), XIV., p. 125, 191 7.
•Nichols and Howes, Phys. Rev. (2), VIII., p. 364 (1916).
« H. L. Howes, Phys. Rev. (2), VI., p. 206, 1915.
* H. L. Howes and D. T. Wilber (2), VII., p. 394, 1916.
•Nichols and Merritt, Phys. Rev. (2), IX, p. 125, Feb., 1917.

Vol. X.!
No. 4. J

FLUORESCENCE OP POUR DOUBLE NITRATES.

349

The mono-ammonium salt, which crystallizes from a solution of the
two component salts in concentrated nitric acid was described by Meyer
& WendeU and crystallographically by Steinmetz.* The crystals are of
the trigonal system with an axial ratio of a : c = i : 1.0027.

The di-ammonium salt crystallizes from a slightly acid water solution
of the two salts in which the ammonium nitrate is in excess of that re-
quired for the mono-ammonium salt. This salt was at first thought to
be the a modification of ammonium uranyl nitrate made by Rimbach*
and measured by Sachs* but an examination of the spectrum of the a
modification so called proved that it was simply uranyl nitrate hexa-
hydrate. The crystals analyzed by Rimbach were probably the mono-
ammonium form as this sometimes forms in the same solution. The
crystals of the di-ammonium salt belong to the monoclinic system. The
mono-potassium salt crystallizes from nitric acid solution in the rhombic
system as described by Steinmetz with axial ratio o : 6 : c = .8541 : i
: .6792.

The di-potassium salt crystallizes with reluctance; but when seeded

UTT

t.
II I i\

""lli iii'l

Li.i

I' ' Mii '

didi

iiii

I'M M ll'l II

mIiIIIii

'Tir'?irii'7i'i'T''iT"rp'''f ii

'i ■ II-' ll'l" ilwi" 'iJi'i ■I'll" II

I "'"I Umll

Jhll I llllh I lilL

M r

JUL

l|Mj||i| II ||i

2400

Fig. 1.

1. Fluorescence and absorption spectra of Mono-ammonium uranyl nitrate. NHfUOt
<NO«)i.

2. Di-ammonium uranyl nitrate, (NH4)jUO»(NO«)4.2H20.

3. Mono-potassium uranyl nitrate. KUOa(NO»)j.

4. Di-potassium uranyl nitrate. K2UOa(NOi)4.

» Meyer and Wendel. Ber. d. d. Ch. Ges., Vol. 36, 4055, 1903;
^Steinmetz, Groth's Chem. Krys., II., p. 150.

* Rimbach, Ber. d. d. Ch. Ges., Vol. 37, 472, 1904.

* Sachs. Zeitschr. f. Krys., Vol. 38, 497, 1904.

350

H, L, HOWES AND D. T. WILBER.

rSSCOND

LSeriss.

from an acid aqueous solution, it forms in beautiful, fluorescent crystals
of the monoclinic system. The axial angle j8 = 90® ± and the axial
ratio a :b :c ^ -6394 : i : .6190. The composition is different from
that of the di-ammonium salt, since it lacks the water of crystallization.

Table I.

Series in the Fluorescence Spectrum of Mono-ammonium Uranyl Nitrate.

l/A.

A(X/A).

l/A.

A(l/A).

l/A.

A(l/A).

1,797.6

88.3

1.573.9

89.1

1,859.1

89.7

A <

1,885.9

86.2

1.663.0

88.0

K-

1.948.8

87.0

1.972.1

86.2

1,751.0

87.2

2,035.8

2,058.3

1,838.2

87.7

1.925.9

87.9

1,602.1

90.6

►

1,6129.4

88.7

2.013.8

87.5

1,692.7

88.2

1,718.1

87.8

2,101.3

1,780.9

88.1

1,805.9

87.5

1.869.0

84.5?

1,893.4

90.7

1,670.0

88.0

1,953.5

87.5

1.984.1

1,758.0

87.7

2,041.0

1.845.7

88.5

1,555.5

89.6

1.934.2

88.1

1,704.2

86.2

1,645.1

88.9

2,022.3

87.9

1,790.4

88.4

1,734.0

87.0

2.110.2

1.878.8

86.1

1,821.0

88.1

1,964.9

85.7

1,909.1

87.8

1.852.5

88.9

2,050.6

1,996.9

89.2

1,941.4

^ 2,086.1

Series in the Absorption Spectrum of Mono^mmonium Uranyl Nitrate.

l/X.

A(l/X).

i/X.

A(l/X).

l/X.

A(l/X).

2,132.8

74.3

2,163.1

74.2

2,111.0

76.7

2,207.1

73.4

2,237.3

75.9 .

•

2.187.7

74.9

2,280.5

75.8

2,313.2

73.7

»^

2.412.5

74.8

2,356.3

73.8

2,386.9

2.562.1

2.430.1

72.4

2,502.5

2,469.1

76.4X2

2,621.9
2,693.9

72.0

Both visual and photographic measurements of the spectra were taken,
and since they agreed well, were averaged together. When possible the
absorption spectrum was obtained by transmitted light. The crystals
from an acid solution were of a deeper green color than those from a water
solution, which necessitated grinding to about .4 mm. thickness to make
them sufficiently transparent. Since the immersion in liquid air spoiled
a crystal many crystals of each form had to be prepared.

Since, as is usual with the uranyl salts, we have in these spectra series

Vol. X.l

Na 4- J

FLUORESCENCE OF FOUR DOUBLE NITRATES,

35^

Table II.

Series in the Fluorescence Spectrum of Di-ammonium XJranyl Nitrate.

I /A.

A(«M).

l/A.

A(l/A).

l/A.

A(l/A).

1,773.6

83.8

1,637.7

84.8

1,664.5

84.2

1,857.4

84.4

1,722.5

84.3

1,748.7

83.5

1,941.8

84.1

1.806.8

84.3

1,832.2

86.0

2,026.4

1,891.1

85.4

1,918.2

86.3

1,976.5

85.4

2,002.5

82.1

1,695.6

83.9

2,061.9

2,084.6

1,779.5

84.9

1,564.0

86.0

1,864.4

84.9

1,650.0

83.3

1,754.1

83.9

1,949.3

1.733.3

83.5

iC-

1,838.0

85.1

G-

1,816.8

84.0

1,923.1

84.9

►

1,786.4

85.2

1,900.8

84.5

2,008.0

1,871.6

83.6

1,985.3

83.2

1,955.2

84.4

2.068.5

1.595.4

85.6

2,039.6

1,681.0

83.4

1,572.9

84.9

1,764.4

83.5

1,628.8

1,657.8

83.8

1,847.9

83.3

1,713.1

84.3

1,741.6

82.9

1,931.2

84.7

1,796.3

83.2

1,824.5

84.0

2,015.9

1,880.8

84.5

1,908.5

84.7

1,965.2

84.4

1,993.2

83.7

2,050.2

85.0

2,076.9

Series in

the Absorption

\ spectrum of Di

-ammonium Uranyl Nitrate.

I /A.

A(I/A).

I /A.

A(l/A).

X/A.

A(l/A).

2,114.8

70.6

2,131.0

70.0

2,092.9

71.0

2,185.4

69.3

2,201.0

67.8

2,163.9

69.3

2,254.7

69.1X4

2,268.8

69.6

2,233.2

70.8

2,531.0

69.3X2

2,338.4

69.9

2,304.0

69.8

2,669.5

2,408.3

69.1

2,373.8

69.8

2,477.4

68.2

2.443.6

67.5

2,178.1

70.6

2,545.6

69.1X2

2,511.1

73.6

2,248.7

72.6

2,683.8

2,584.7

72.8

2,321.3

71.3

2,140.8

70.1

2,657.5

2,392.6

2.210.9

68.2

2,102.3

71.6

2,124.5

2,279.1

2,173.9

71.2

2,332.5

69.3X3

2,077.3

70.9

2,245.1

71.7

2,401.3

68.8

2,148.2

69.8

2,316.8

2,472.2

70.9

2,218.0

73.0

2,384.8

69.4

•

2,291.0

67.5

2,454.2

69.1

2,344.9
2,414.9

70.0
69.8

2,358.5
2,429.2

70.7
68.5

2,523.3
2,592.7

69.4

2,484.7

68.1

2,497.7

69.3

■

2,552.8

68.4

2,567.0

2,422.3

68.6

2,621.2

2,154.2

70.1

h-^

2,490.9

68.8

2,224.3

70.6X2

2,559.7

68.4

•

2,365.4

70.7

^•

2,628.1

2,436.1

68.1

2,538.1

68.1

2,504.2

2,602.2

352

H. L. HOWES AND D. T. WILBER,

[Sbcond

of constant frequency intervals, the tables contain the frequencies of the
bands, where the frequency unit is of such that for X = 5,000 A.U. the
table reading is i/X = 2,000 units. The average positions given should
never be in error more than two frequency units.

The relation between fluorescence and absorption series is of the same
nature as that previously found to exist in the spectra of the uranyl
salts. Fig. I indicates the four spectra.

By referring to Table I. fluorescence series '* /" will be seen to
consist of six bands beginning with the red band at 1,670.0 fre-

Table III.

Series in the Fluorescence Spectrum of Mono-potassium Uranyl Nitrate.

x/X.

A(i/X).

88.5

X/X.

A(iA).

x/X.

A(xA).

1.725.3

1,589.8

85.0

1,615.8

84.6

1,813.8

86.6

1.674.8

87.2

1.700.4?

89.8

1,900.4

87.7

1.762.0

86.5

1.790.2

87.3

1,988.1

1,848.5

86.4

1.877.5

87.0

1.934.9

86.5

1.964.5

85.8

1.569.1

86.8

2.021.4

86.4

2.050.3

86.0

1,655.9

86.8

• V

2.107.8

2.136.3

1,742.7

87.5

D-

1,830.2

86.1

1.683.2

86.2

1,916.3

87.5

1.867.6

87.5

1.769.4

87.5

2,003.8

86.7

1.955.1

88.1

1.856.9

86.8

2.090.5

2.043.2

j["

1.943.7

86.8

2.030.5

1.754.1

88.2

2.118.2

1.842.3
1.928.5

86.2
87.2

2.015.7

Series in the A bsorption Spectrum of Mono-potassium Uranyl Nitrate.

i/X.

A(i/X).

d-i

2,167.0
2,238.3
2,313.8
2.386.1

71.3
75.5
72.3

x/X.

2.117.7
2,189.6

A(x/X).

71.9

x/X.

2.140.3
2.213.2
2,287.9?

A(x/X).

72.9
74.7

quency units and continuing to the blue band at 2,110.2. Ab-
sorption series **i*' in Table I. has the first member at 2,111.0
which almost agrees in position with the last fluorescence band, of
series '*/*', but the intervals between the absorption bands are seen
to be shorter than those between fluorescence bands. The nomen-
clature of the fluorescence and absorption series is so chosen that the
series which are related have the same letter, e. g., " A ** is related in the

VOL.X.1

No. 4. J

FLUORESCENCE OF FOUR DOUBLE NITRATES.

353

same manner as above to *' a," '* J3 *' to " 6," etc., although the ** reversing **
band is not generally present. Consider series ** A *' for example, which
ends with band 2,058.3; the first absorption band of series '* a " is not
at 2,058.3 but at 2,132.8. If we note that the- absorption interval is

Table IV.

Series of the Fluorescence Spectrum of Di-potassium Uranyl Nitrate,

x/X.

A(xA).

x/X.

A(x/X).

X/X.

A(x/X).

■

1,775.2

86.7

1,723.8?

84.8

1,651.5

86.1

1,861.9

87.0

1,808.6

85.5

1,737.6

86.2

1,948.9

85.7

F-

1,894.1

86.2

1,823.8

87.4

2,034.6

1,980.3

88.5

1,911.2

87.3

2,068.8

1,998.5

87.0

1,621.3

87.7

2.085.5

1,708.0

85.7

1,554.0

86.4

D-

1,793.7

86.3

1,640.4

87.2

1,831.8

87.7

1,880.0

86.9

1,727.6

85.6

1,919.5

87.5

1,966.9

86.8

1,813.2

85.8

2,007.0

2,053.7

1,899.0

87.2

1,986.2

1,663.3

88.5

1,631.6

86.3

1,751.8

86.0

1,717.9

84.8

f 1,903.7
1 1,906.8

86.5

1,837.8

87.3

£-

1,802.7

86.6

1,925.1

86.8

1,889.3

86.5

f 1,989.7
1 1.993.8

2,011.9

1,975.8

86.4

Za ^

2,062.2

f 2,075.8
1 2,080.7

86.6

1,670.8

87.0

1,757.8

86.8

L-

1,844.6
1,931.4

86.8
86.9

•

2,018.3

Series in the Absorption Spectrum of Di-potassium Uranyl Nitrate,

x/X.

A(x/X).

x/X.

A(x/X).

x/X.

A(x/X).

2,196.6

72.7

2,152.6

71.7

2,105.0

74.9

2,269.3

2,224.3

2,179.9

2,210.9

74.8

2,169.2

70.9

2,253.4

71.7

2,285.7

2,240.1

70.8

2,325.1

71.8

2,310.9

71.7

2,396.9

2,369.4

75.0

2,382.6

2,444.4

2,361.6

76.2

2,437.8

75.4

•._

2,513.2

about 74 units and add 74 to 2,058.3 we obtain 2,132.3, which is the
reason for classifying series ** a ** as we have.

In Table II. there are three extra absorption series which are not so
intimately related to fluorescence series, hence they are designated by

354

H. L, HOWES AND D. T, WILBER.

[Second
r

the Greek letters. This is also true of series " /' " and " B " in Table
IV. In the tables of series of the mono-ammonium and mono-potassium
salts there are not sufficient absorption series to match with fluorescence
series, but there are no extra absorption series which do not join. In the

Table V.

Average Intervals.
Mono-ammonium Uranyl Nitrate.

Fluorescence series

Absorption series

Ratio of fluorescence to absorption

86.6

88.3

87.7

73.7

74.5

74.2

A/a

Did

Gig

1.18

1.19

1.18

88.1
%

75.1

1.17

Di-ammonium Uranyl Nitrate.

Fluorescence series

Absorption series

Ratio of fluorescence to absorp-
tion

A
84.4

a
69.2

Ala
1.22

B
84.4

h
71.6

BIh
1.18

84.3

c
69.8

CIc
1.21

D
84.5

d
68.9

Did
1.23

E
85.0

e
68.7

Ele
1.22

83.7

g
68.8

Gig
1.22

/
83.9

69.6

1.21

/
83.8

J
69.7

Jlj
1.20

84.8

70.9

84.0
71.5
Lll

1.201 1.18

Mono-potassium Uranyl Nitrate.

Fluorescence series

Absorption series

Ratio of fluorescence to absorption

K
86.6

k
74.1

KIk
1.17

Di-potassium Uranyl Nitrate,

Fluorescence series

Absorption series

Ratio of fluorescence to absorption

D
86.6

d
72.7
Did
1.19

E
86.2

e
74.8

Ele
1.16

87.2

/
75.0

H
86.5

h
71.7

1.16 1.21

K
86.9

71.3
KIk
1.22

L
86.9

/
74.9
Lll
1.16

di-ammonium spectrum there is a related absorption series for each
fluorescence series.

The study of related series is made confusing by the presence of ab-
sorption series extending into the fluorescence region; and, vice versa,
many fluorescence series, if extended, fit absorption bands. A special

VOL.X.

Na4.

]

FLUORESCENCE OP FOUR DOUBLE NITRATES.

355

Study of the over-lapping region is needed to determine better the relation
between fluorescence and absorption series. Such a study is now in
progress. As to the completeness of classification of bands into series it
can be said that not a fluorescence or absorption band of any of the salts
fails to fit into one of the constant frequency series.

In the study of the spectrum of the uranyl nitrate by Merritt* each
fluorescence series has constant intervals between the bands which are
the same for all series; in the case of the absorption series, however, the
interval is not the same for all series. In our study of the double nitrates
we find an unmistakable variation in the fluorescence intervals as well

IfOO

A
I

T
•

I I I

J
I

K
I

I I

' ■ ' ■

A
-L.

J K,

L M

t I I

I I

■

I I

" ' ■ ' '

Fig. 2.

A single group from each of the four spectra.

1. Mono-potassium uranyl nitrate. — Trigonal.

2. Di-potassium uranyl nitrate. — Monoclinic.

3. Mono-ammonium uranyl nitrate. — Rhombic.

4. Di-ammonium uranyl nitrate. — Monoclinic.

The bands occupy their natural positions in the left-hand panel, but have their strongest
bands in vertical alignment in the right-hand panel.

as in the absorption. In the mono-ammonium nitrate the interval
varies from 86.4 for series '* M ** to 89.0 for series " B.** In the di-am-
monium nitrate the interval varies from 83.7 for ** G " to 85.0 for '* E.**
In the mono-potassium nitrate spectrum the interval varies between
86.4 for " G " and 87.9 for *' /.*' In the di-potassium nitrate the interval
varies between 86.2 for '* E " and 87.6 for ** /."

» Nichols and Merritt, Phys. Rev. (2), Vol. IX., p. 125, Feb.. 1917.

356 H. L. HOWES AND D, T, WILBER, [^SS

The variation of the interval in the absorption series is of the same
order of magnitude; e. g., an extreme variation of 1.4 in the mono-am-
monium, 3.5 in the di-ammonium, 4.3 in the mono-potassium and 4.4 in
the di-potassium nitrate. In this connection it was thought to be of
interest to compare the ratios of related fluorescence and absorption
intervals. In Table V. these ratios are given. The ratios are nearly
constant for the mono-ammonium and mono-potassium uranyl nitrates;
but differ in the case of the other two salts.

That the crystal system to which a salt belongs is an important factor
in determining the position of the bands can be seen in Fig. 2. In the
left-hand panel a single group is shown in its natural position; in the right-
hand panel the strongest bands of each group are placed in the same
vertical line, to show the resemblance in grouping. This grouping is
probably due to the fact that all four belong to the same chemical family.
If we compare this grouping with that of the uranyl nitrate spectra studied
by Merritt we find little resemblance, hence the grouping is probably
characteristic of the double uranyl nitrate family. In the left-hand panel
it will be seen that the second and fourth groups occupy almost identical
positions, while the first and third occupy positions which differ from one
another, and from the second or fourth. As has previously been stated
the second and fourth groups belong to the monoclinic crystal systems,
the first to the trigonal and the third to the rhombic system. Since all
four spectra vary slightly in their frequency intervals the relative posi-
tions would change slightly if we compared homologous groups in the
other end of the spectrum, but this gradual and slight shifting would
not change the general condition which indicates that the absolute posi-
tion of a group is largely determined by the crystal system. This is not
entirely new, as the four triclinic crystals of the double uranyl chlorides
studied by Nichols and Howes exhibited spectra which were as nearly
coincident as could be expected of salts which vary in molecular weight.

Again, in the case of the uranyl nitrate, the crystals of the hexahydrate
were of the rhombic system, while those of the trihydrate and dihydrate
were of the triclinic system. In spite of slight shifts due to changing
molecular weight the strong bands of the two spectra produced by the
crystals of the triclinic system agree fairly well, while the strong bands
of the spectrum produced by the rhombic crystal reside in entirely
different positions.

There is one more bit of evidence which adds weight to the above view.
The chemical formulae of the two potassium salts are more nearly alike
than those of the two ammonium salts since the di-ammonium salt has
two molecules of water of crystallization, while the other salts have none,

Na"i^*] FLUORESCENCE OF FOUR DOUBLE NITRATES, 357

yet there is a greater difference between the third and fourth spectra than
there is between the first and second spectra.

Summary.

1. The spectra of the double uranyl nitrates resemble those of the
previously studied uranyl salts in that the bands can be arranged in series
having constant frequency intervals.

2. These intervals while constant for any given series are different for
different series.

3. In the mono-ammonium uranyl nitrate and the mono-potassium
uranyl nitrate the ratio of the interval of a fluorescence series to the
interval of the absorption series which joins that fluorescence series is
approximately a constant.

4. Although the grouping of the bands shows a strong family resem-
blance in the four spectra yet the absolute position of a group is largely
determined by the crystal system.

Physical Laboratory of Cornell UNrvBRSiTY,
April 6. 1917.

358

ALPHEUS W. SMITH.

[Sbcomd
i

THE REVERSAL OF THE HALL EFFECT IN ALLOYS.

By Alpheus W. Smith.

TN studying the Hall effect in bismuth-tin alloys von Ettingshausen and
-■- Nernst^ made the interesting observation that for a certain strength
of the magnetic field there is a reversal in the direction of the Hall elec-
tromotive force. For small values of the magnetic field the direction of
this electromotive force is the same as that in bismuth. As the magnetic
field is increased the Hall electromotive force at first increases, passes
through a maximum, and sinks to zero, after which it reverses its direction
and increases continuously. To account for this reversal von Ettings-
hausen and Nemst put forth the following suggestion which they did
not undertake to verify^

In Fig. I let the plate AB have a current of electricity flowing from A
to B. If there is a magnetic field perpendicular to the plane of the plate,
two phenomena are observed simultaneously. If the direction of the
flow of the Amperian magnetizing current is that indicated by the arrow
on the circle, the upper edge of the plate becomes negative with respect

to the lower and its temperature is increased
above that of the lower. There is thus set
up a temperature gradient from the top to
the bottom of the plate. If the plate is per-
fectly insulated the flow of heat produced by
this temperature gradient is just balanced by
the heat generated at the top in excess of
that at the bottom of the plate. The difference of temperature thus
established is a measure of the von Ettingshausen effect and is given
by the equation

^ Hi

where i is the current in absolute units, H the magnetic field and d
the thickness of the plate in centimeters. The Hall electromotive force
is given by the equation.

Fig. 1.

E =

T —

* Ann. der Phys., 33, p. 474, 1888.

NoT^*] REVERSAL OF HALL EFFECT IN ALLOYS. 359

If it happens that the plate is imperfectly insulated from its surround-
ings there will be a flow of heat from the medium into the plate. This
flow will enter the bottom where the plate is colder than the surroundings
and leave it at the top where it is warmer than the surroundings. It
seems to me that a measure of the maximum flow to be expected from
this source may be had by regarding the actual temperature gradient in
the plate reversed in direction, so that there is in the plate from bottom
to top a flow of heat in the direction opposite to that which would result
from the observed temperature gradient in the plate. This flow is in-
dicated by the vertical arrow in Fig. i.

It has been observed in these same alloys that if the flow of electricity
is replaced by a flow of heat as indicated in Fig. 2 and if the thermo-

electromotive force between two points A and B on the longitudinal axis

■

of the plate is compensated, under the magnetic action there is set up a

difference of potential between these points.

Of course this potential difference may be

interpreted as a change in thermo-electro- h j ( >i 91 u^

motive force on account of the magnetic

action. This difference of potential is not

reversed with the magnetic field. In bis- _..

muth and these alloys it is found that its

direction is such that the hot end of the plate is positive with respect

to the cold end. The magnitude of this potential difference is given

by the equation,

where dt/dx is the temperature gradient in the plate.

Now suppose that on account of the transverse flow of heat which von
Ettingshausen and Nernst assume to be associated with the Hall effect,
there is established this longitudinal effect between the top and the bottom
of the plate. This would give an electromotive force which would be
superposed on the ordinary Hall electromotive force. These two electro-
motive forces would be opposite in direction.

The observed electromotive force would be their algebraic sum. Let
E = the Hall electromotive force between the top and bottom of a plate
one centimeter in thickness when a current of one electromagnetic unit
is flowing in it. Let At = the corresponding difference of temperature
between the top and bottom of a plate one centimeter wide. Let £2
= the longitudinal potential difference established in the plate by the
magnetic action when the temperature gradient is one degree per cm.

360 ALPHEUS W. SMITH, [^SS

Let El = observed Hall electromotive force. Then,

£1 = £ - EiAL

Whether this will account for the reversal is obviously a question of the
relative magnitudes of these potential differences.

More recently Becquerel^ observed that it is possible to obtain this
reversal of the Hall electromotive force in a plate cut from a crystal of
bismuth. In such plates the magnetic field at which the reversal takes
place depends on the direction of the crystalline axis with respect to the
magnetic field and the plane of the plate. Chapman* has observed that
the Corbino effect is reversed in these bismuth-tin alloys under essentially
the same conditions under which the Hall electromotive force is reversed.
In the Corbino effect the plates are discs. The current enters at the
center of the disc, flows along the radius and leaves at the periphery.
The magnetic action causes the current to have a component normal to
the radius of the disc. The effective electromotive force producing this
component of the current is essentially a Hall electromotive force. Under
the conditions under which it is produced it is not clear that the tem-
perature effect to which von Ettingshausen and Nemst attribute the
reversal can arise.

The author has observed that the Hall effect is reversed in bismuth-tin
alloys for alternating currents with a frequency of 60 cycles per sec. It
would seem that for alternating currents of this frequency the accompany-
ing thermal effects on which the explanation of von Ettingshausen and
Nernst is based, would scarcely have sufficient time to become effective.
The fact that the reversal occurs under these conditions shows that either
the explanation is not to be ascribed to the accompanying thermal effects
or that these thermal effects manifest themselves more quickly than is
ordinarily supposed.

In view of these considerations it seemed worth while to study the
reversal of the Hall effect in these alloys, extending the observations to
higher field than those used by von Ettingshausen and Nernst. The
reversal has also been observed in alloys of bismuth and lead. The
principal point of interest is to see whether this reversal of the Hall effect
as ordinarily defined can be caused by the superposition of some of the
allied phenomena on the Hall electromotive force.

The continuous curves in Fig. 3 which have been plotted from the
observations of von Ettinghausen and Nernst show the relation between
the observed Hall electromotive force and the magnetic field for two

^ Comp. Rendus, 154, pp. 1795-1798, 1912.
» Phil. Mag., 32, pp. 303-326, 1916.

Vol. X.!
No. 4. J

REVERSAL OF HALL EFFECT IN ALLOYS.

361

-2

5°

+1
+2

alloys of bismuth and tin. The ordinates are the Hall electromotive
forces in a plate one cm. thick with a current of one electromagnetic unit
in it. One of these alloys contained 0.95 per cent, tin; the other con-
tained 1.46 per cent. tin. The dotted curves in these figures show the
potential difference to be expected from the thermal flow assumed by
von Ettingshausen and Nernst. It must be remembered that the
measure of this flow of heat is obtained by assuming the temperature
difference between the top and bottom of the plate, arising from the
magnetic action on the electric current, is reversed. This flow is then at
right angles to the direction in which the electric current is flowing. On
account of this flow of heat the magnetic field produces a difference of
potential between the top and bottom of the plate which would be either
added to, or subtracted from the Hall electromotive force. The or-
dinates for these curves are ob-
tained by multiplying the longi-
tudinal potential difference for a
particular magnetic field when
the temperature gradient is one
degree per cm. in the plate by
the difference of temperature be-
tween the top and the bottom
of the plate due to the Ettings-
hausen effect when the plate is ^ 0
one centimeter wide and one |I$+2
centimeter thick and is travers-
ed by a current of one electro-
magnetic unit. It is obvious
from these curves that these longitudinal potential differences are much
too small to account for the observed reversal of the Hall electromotive
force.

Since the properties of tin and lead are very similar it seemed probable
that alloys of bismuth and lead would show the reversal of the Hall
electromotive force which had been observed by von Ettingshausen and
Nernst in alloys of bismuth and tin. In order to test this, three alloys
of bismuth and lead were prepared by fusing together in known pro-
portions by weight Baker's analyzed bismuth and Kahlbdum's pure
lead. These alloys were then cast in the form of rectangular plates about
0.1 cm. thick, 1.5 cm. wide, and 4.0 cm. long, with narrow arms pro-
jecting from the middle of each of the longer sides of the rectangle.
These arms were sufficiently long to project outside of the more intense
part of the magnetic field. To these arms were soldered the lead wires

-4-

-2

•

--.

IT.

. - ?v

• • ■

•»«- "■

~ "

•

::^

ofj-j

■^

«^i«H

• • «

" *■ •

• ---

- .i>_

—

. -^

s
Fig. 3.

10 HtW

362

ALPHEUS W, SMITH.

rSSCOND

lSbribs.

which were joined to the galvanometer on which the Hall electromotive
force was observed. By filing these arms near the edge of the plate
they could be shifted until they were nearly on the same equipotential
line. Heavy strips of copper were soldered along the ends of the plate
and served as electrodes by which the current entered or left the plate.
Care was taken that the lines of flow be as nearly as possible parallel to
the edges of the plate. With such a plate it is unnecessary to make cor-
rection for the thermo-electromotive force arising from the temperature
difference between the top and bottom of the plate set up by the magnetic
action, for the lead wires where they are joined to the plate are of the same
material as the plate and the temperature difference which results from
the magnetic action will not produce an appreciable thermo-electro-
motive force. Essentially no temperature change occurs where the
arms are soldered to the lead wires. This form of plate avoids the
necessity of making that correction for the Ettingshausen effect which is
necessary where the lead wires are soldered directly to the plate.

For further details with re-
spect to the measurement of
the Hall effect in these alloys
reference is made to an earlier
paper on **The Variation of
The Hall Effect in Metals with
Change of Temperature.*'^
Except for minor details the ar-
rangement of the apparatus and
the method of taking observa-
tions were the same as used in
that paper.

In order to know whether the
idea suggested by von Ettings-
hausen and Nemst would ac-
count for the reversal of the
Hall effect which was observed
in these alloys of bismuth and
lead, it was necessary to know
the Ettingshausen effect and the
longitudinal potential difference
in them. Observations had already been made on the Ettingshausen effect
in these alloys in connection with the study of this effect in several series
of alloys. A description of the method used and of the results obtained

^^^^^

^^^^

•

»2i

)

-20

-iZ

-©-T

•

^ M

"" ^

■ ^ ■

• -

+8
+i2

+20
+2f

20 Hxitf*

Fig. 4.

» Phys. Rbv., 30, p. I, ipio.

Vol, X.1
Na4. J

REVERSAL OF HALL EFFECT IN ALLOYS,

363

in that investigation were published in the Physical Review, N. S.,
Vol. VIII., p. 82, 1916. The necessary data for the present purpose were
taken from the results given in that paper.

For the investigation of the longitudinal thermomagnetic potential
difference the same method was used which the author has previously
described in the Physical Review, N. S., Vol. II., p. 383, 1913. In the
present investigation no essential departures from the details of that
method were made.

The curves showing the results obtained in the study of these three
alloys of bismuth and lead have been given in Figs. 4, 5 and 6. These

-s

-2
O

■»■«
♦10

>•

i2jti

■#c;

. - .

■

"■

-16

-12

-8

-t

+g
\X0

ZO H xi'd^

«0

<r#

rA^

■^

....

^ ^ "

• " ■■

••• -

■^v

ZO HxlO*

Fig. 5.

Fig. 6.

alloys contain 3 per cent., 5 per cent., and 10 per cent, of lead respectively.
From the observations on the Ettingshausen eflfect and on the longi-
tudinal potential difference, the potential differences to be expected
from the transverse flow of heat assumed by Ettingshausen and Nernst,
have been calculated. These potential differences have been plotted for
ordinates in the dotted curves given in Figs. 4, 5 and 6. Here as in the
alloys studied by von Ettingshausen and Nernst the potential differences
which might arise from the longitudinal potential effect are much too
small to account for the reversal of the Hall eflfect.

Becquerel has suggested that such curves, showing the relation between
the Hall electromotive forces and the magnetic fields may be split up into
two curves, — Curve -4, a straight line passing through the origin and
Curve B which arises to a fixed value after which it is parallel to the
horizontal axis. The sum of the ordinates of these two curves gives the

364

ALPHEUS W, SMITH,

[Second
Seribs.

ordinates of the observed curve. This analysis regards the Hall effect
as made up of two parts which are opposite in sign. Of course this
analysis is arbitrary, as a number of other pairs of curves may be chosen.
It is, however, suggested by the fact that for the larger values of the
magnetic field the Hall electromotive force is proportional to the magnetic
field. Furthermore, this analysis is helpful in the present discussion
because it shows the least correction that must be added to the Hall
electromotive force to account for the observed reversal. The slope of
Curve A is more than forty times that of the corresponding dotted curve
showing the possible correction arising from the longitudinal potential
difference. Even admitting the validity of the assumption made by von
Ettinghausen and Nemst concerning the reversed flow of heat it is
necessary to conclude that their suggestion cannot account for the reversal
of the Hall effect.

In Fig. 7 the Hall electromotive forces in bismuth and in three alloys

^
^

U3
-8

•4.

■

1
k

-a

;

+4-

20 HXlfl*

— /

-8

—A.

-2

Si=

♦s

•-

+4-

•^

eo H*io"

Fig. 7.

Fig. 8.

of bismuth and lead have been plotted against the magnetic fields pro-
ducing them. In Fig. 8, similar curves are shown for bismuth and two
alloys of bismuth- and tin. The characteristics of each of these sets of
curves are essentially the same. Small additions of either lead or tin to
bismuth causes a rapid decrease in the Hall effect. For any one of the
alloys the Hall electromotive force at first has the same direction as in

Na'4^'] REVERSAL OF HALL EFFECT IN ALLOYS. 365

bismuth. With increasing field it increases to a maximum, reverses its
direction and then increases proportional to the magnetic field.

It is necessary to conclude from these and other observations that there
has yet been no satisfactory suggestion advanced to account for the
reversal of the Hall effect under the conditions considered in this paper.
The explanation of this interesting fact is probably to be looked for in the
structure of the alloy, particularly in the interstices between the vibrating
atoms.

Physical Laboratory,

Ohio State University.

366 C. W HEAPS,

rSECOWB

RESISTANCE AND MAGNETIZATION.

By C. W. Heaps.

IN its present condition the theory of free electrons in metals explains
qualitatively the general aspects of most of the observed phenomena,
but fails to account quantitatively for some of the more special experi-
ments. In particular, the Hall effect and the other galvanomagnetic
phenomena exhibit peculiarities in different metals which are difficult
to bring into line with the theory. The difficulty appears to arise because
of complex and little understood molecular actions inside the metal
which the theory does not take into account. For this reason the above
mentioned class of phenomena deserves special consideration. Further-
more, the explaining of the motion of free electrons in magnetized metals
involves the explanation of magnetostriction and of magnetism. The
results of some previous experiments* made by the writer have indicated
very clearly that the resistance change induced in iron and nickel by
magnetization is very intimately associated with the structural changes
producing magnetostriction. In order to bring out this relationship
more clearly it seemed important to make experiments on the influence
of structure on the resistance change of metallic conductors when placed
in magnetic fields. The present paper describes these experiments.

In interpreting the results it will be necessary to consider the theoretical
side of the question. On the usual theory of free electrons in a metal
it is found that when a magnetic field acts at right angles to an electric
current in the metal the expression for the change of resistance involves
the term IP(e/myT^ multiplied by a numerical factor. Here H is the
magnetic field and T the free period of an electron. The numerical
factor by which the above expression is multiplied has been determined
as i * and as -^^ *, and there has been some disagreement as to whether or
not the theoretical resistance change is an increase or a decrease.

In order to clarify our ideas it will be of advantage to consider the
separate factors which enter to change the magnitude of the electric
current in a wire when this wire is magnetized. It is ordinarily assumed
that free electrons in a metal move about with high velocity very much

» Phys. Rev., VI., p. 34. 1915.

* E. P. Adams, Phys. Rev.. XXIV., p. 428, 1907.

»0. W. Richardson, ''The Electron Theory of Matter."

No'ri^'] RESISTANCE AND MAGNETIZATION. 367

like gas molecules but collide with molecules of metal instead of with
each other. When an electric field acts on these electrons a drift motion
is superposed on the ordinary velocity of agitation of the electrons and
it is this drift motion which constitutes the electric current. It has been
shown* that the velocity with which electrons drift under these conditions
is given by the equation

where Uo is the drift velocity in the direction of the electric force X,
Xo is the mean free path of the electrons, and V is the velocity of agitation
of the electrons.

We have now to consider the effect of a magnetic field on this drift
velocity in two special cases as follows: (i) when H is applied parallel
to the drift velocity (the longitudinal effect), and (2) when H is applied
perpendicular to the drift velocity (the transverse effect). In case (i)
it is clear that the only way in which H can change Uo is by affecting the
mean free path of an electron. Such an effect of H on Xo is to be expected,
for the existence of magnetostriction implies an alteration of inter-
molecular distances and a consequent change in Xo. It has also been
pointed out* that the paths made by electrons between collisions would
on the average be curved in the case where a magnetic field acts, and
hence would be changed in length for this reason also. If X* is the mean
free path in the longitudinal magnetic field we may write X* — Xo = 5X

8U 8\

l7o = x"o' ^'^

where dU is the increase in Uo produced by the increase 8\ in Xo. The
electric current, /, is proportional to Uo in case the magnetic field does
not affect the number of electrons per unit volume, so that if 5/ repre-
sents the increase of current produced by S\ we have

81 _SX

I " Xo*

For the increase of resistance we may write

6R d\

R^'To' * (3)

According to this equation a longitudinal magnetic field will produce
an increase of resistance when the mean free path of the electrons is
decreased by H.

* Langevin, Ann. de China, et de Phys., 28, p. 336, 1903.
« Richardson. "Electron Theory of Matter."

368 c. w. HEAPS, [iSSJ!

In case (2) we have an added effect to consider. Townsend has
shown^ that if Xo is unchanged by H the drift velocity in a transverse
magnetic field is given by

where (a = H(e/m), and T, the mean value of the time between collisions,
is equal to Xo/F. We have seen, however, that a magnetic field may be
expected to change Xo. If we let X* — Xo = dX be this change produced
in Xo by a transverse magnetic field we must use X* instead of Xo in
equation (4). We thus have, approximately,

Unii + i^T') = C/or»

dU U,-Uo d\

The change of electric current is proportional to the change in £7o so
that this expression gives the increase of current produced by a transverse
magnetic field. For the increase of resistance we may write

dR ^^ d\

According to this equation a transverse magnetic field produces an
increase of resistance unless d\ is an increase in free path sufficiently
large to make dX/Xo greater than w^T^.

If we subtract equation (3) from equation (6) we get

---^Ipy^j--.- + -. (7)

In the general case we may not set the last two terms of this equation
equal to each other because of the crystalline nature of the specimens
under examination. It is usual to experiment with wires which have
been pulled through draw-plates, and in specimens of this kind it is
possible for the crystal structure to differ along different directions in
the metal. A magnetic field perpendicular to the wire, therefore, might
produce an effect on molecular arrangement which is different from the
effect of a field parallel to the wire. In case we have a substance which
is magnetically isotropic the last two terms of the equation above become
equal and we get

R R -^U P- ^^>

This equation has the advantage of being free from terms involving

» "Electricity in Gases."

J5S"^] RESISTANCE AND MAGNETIZATION. 369

unknown molecular changes and hence may be used for determining T,
If we take the electrical conductivity of a metal to be given by^

2 ^Xo

<T = * ft 77»

i. mV

where n is the number of electrons per unit volume, we get

dR 6R 9^„ a*

According to this equation a transverse magnetic field should always
produce a greater increase of resistance than a longitudinal field.

Experiments.

It has been observed* that some forms of graphite suffer a large re-
sistance change while other forms are apparently unaffected by a magnetic
field. No systematic study seems to have been made of the reasons for
these variations, and inasmuch as the variations are quite large it seemed
as if a knowledge of the causes of these variations would prove illumina-
ing from the standpoint of the theory. Furthermore, powdered graphite,
composed of small crystals, may be compressed into bars, and these
bars are similar to most metals in that they are composed of crystal
agglomerations.

The apparatus which was used in measuring the resistance of the speci-
mens was a Wheatstone bridge with balancing shunts so arranged that
changing one of these shunt resistances by a large amount compensated
for a very small change in the resistance of the specimen. A Leeds and
Northrup high sensibility galvanometer of resistance 13 ohms was used,
and the apparatus was sufficiently sensitive in most cases to measure
a value of dR/R as small as 3 X io~*. The specimen under examination
was suspended by means of a wooden clamp or an ebonite rod between
the poles of a Weiss electromagnet in such a way that a simple rotation
of the magnet sufficed to change a transverse into a longitudinal field
without disturbing the specimen. This arrangement was found to be
of distinct advantage, since the moving or jarring of the specimen was
found under some conditions to alter its resistance. The experiments
were performed at room temperature (about 26® C.) and in order to shield
the specimen from air currents thick pads of hair-felt were set up around
it between the magnet poles, which were 2.3 cm. apart. The bridge
current was allowed to flow for a sufficient length of time before taking

> Swann, Phil. Mag., March, p. 441, 1914.

* D. E. Roberta. Konink. Akad. Wetenach. Amsterdam. Proc. 15, p. 148. 191 2. G. E.
Washburn. Ann. d. Phys.. 48. 2. p. 236. 1915.

370

C. W. HEAPS.

[

Second

measurements to let the specimen come into temperature equilibrium
with its surroundings. The resistance of the wires leading to the speci-
men was carefully measured and allowed for in the computations. These
leads, however, were of copper and they had a low resistance, so that the
effect of the magnetic field on the resistance of these leads was negligible.
Several forms of graphite were used in preparing the specimens for
examination. The first group of experiments was made on the graphite
of ordinary pencils, electric light carbons, and rods constructed for
lubricating purposes. In these materials the pure graphite is mixed with
" binder " of some sort — ^usually clay in the case of pencils — so that the
resulting mixture is quite hard and brittle. In all these substances the

effect of a magnetic field was quite
small. Fig. i shows the relation be-
tween H and the resistance change
for a rod of soft graphite made by
the Dixon Crucible Co. for lubricat-
ing purposes. These curves are typi-
cal of other curves obtained for other
materials of this class. It will be ob-
served that the transverse effect is
greater than the longitudinal effect
for a given value of H, The experi-
ments seemed to indicate that in soft
specimens, presumably containing a
small amount of binder, the magnetic field produces a greater effect than
in the hard specimens.

The next group of experiments was made on graphite powder com-
pressed into sticks and bars. The following methods of compressing the
powder were used:

(a) Hand compression between brass electrodes in small bore glass
tubes. The electrodes were held by a screw arrangement in tight contact
with the graphite and the graphite was not removed from the tube during
the experiments.

(&) Compression of the powder by means of a hydraulic press into
grooves in an ebonite block. The groove for each specimen consisted of
a single straight shallow trough connecting two deep holes in the face of
the block. Copper wires coiled in these holes were led out to serve as
terminals. This block was placed face upwards at the bottom of a hollow
iron cylinder, graphite poured into the cylinder, and pressure applied.
When the block was removed and the excess graphite scraped off, the
groove was found to contain a bar of graphite formed by pressure applied
in a direction perpendicular to its length.

Fig. 1.

JJgJ-^-] RESISTANCE AND MAGNETIZATION. 37 I

(c) Compression into the form of thin plates by means of the hydraulic
press. The same method was used as in (b) except that no grooves were
cut in the ebonite block. The thin brittle plates which were formed were
cut into bars with a razor blade, the ends copper plated, and copper leads
were soldered to the specimen.

Two different grades of powder were used — the visible difference
between the two grades being that one consisted of larger crystals than
the other. It was evident upon inspection of the specimens prepared in
the three ways described above that the result was not even approxi-
mately an isotropic material. The bars exhibited cleavage planes per-
pendicular to the direction in which pressure had been applied. We
should, however, expect a result of this nature. The crystals of the
powder were like very small thin plates, hence pressure applied to such
an aggregation might be expected to produce an anisotropic effect by
forcing the crystals to turn their large plane faces perpendicular to the
direction in which pressure had been applied while the specimen was
being made up. When two test specimens were made up according to
methods (b) and (c) respectively, it was found that a magnetic field pro-
duced the greater effect on the sample made by method (c). Since this
method produces a much greater compression of the graphite powder
than the other two methods it appears that high compression tends to
increase the resistance change in a magnetic field. However, it was
possible to change by hand the compression of samples made by method
(a) until the resistance was halved, without producing an appreciable
increase in the effect of a magnetic field. We must conclude, therefore,
that a high degree of compression is necessary in order to produce much of
an effect.

In Fig. 2 the results obtained with the fine power are expressed in
graphical form; in Fig. 3 some of the corresponding results for the coarse
powder are plotted. The letter on each curve specifies the method (de-
scribed above) used in preparing the specimen. Circles are used to
specify all results obtained when the magnetic field was perpendicular
to the direction of the current; small crosses are used when J? was parallel
to the current. When the magnetic field was parallel to the direction
in which pressure had been applied during the process of making the
specimen the symbol || is written on the curve; when i? was perpendicular
to this direction the symbol X is used. The immediate conclusions
which we may draw from these curves are as follows :

I. Increasing the size of the graphite particles increases the effect of
the magnetic field on the resistance; for the curves of Fig. 3 are all higher
than the corresponding curves of Fig. 2.

372

C. W, HEAPS,

[Second
LSbues.

2. The direction of the magnetic field with respect to the crystal
structure in graphite is of greater importance than its direction with
respect to the electric current. For in (b) of Fig. 2 the effect of a trans-
verse field is only a small amount greater than the effect of a longitudinal
field, provided the field is' kept perpendicular to the direction of compres-

•

f'^

>>*■

W -^

'— 5

f.rf

«i

o1^

2fl

.^^

/■

<P^

■^

*T3»

id»

iWO

Fig. 2.

Fig. 3.

sion. Turning the field parallel to the direction of compression increases
the effect of the transverse field.

3. If the effect of crystal structure in the graphite is eliminated we
secure an effect in a transverse field which is greater than in a longitudinal
field. This conclusion follows if we assume magnetic isotropy in all
directions perpendicular to the direction of compression. We are then
able to compare the two lowest curves of Fig. 2 with each other on the
assumption that the influence of crystal structure is the same for each
curve. In Fig. 2 the transverse field of 8,000 gausses gives a value of
dRfR, which is greater by about 8 X io~* than that of the longitudinal
field.

In order to test these conclusions more completely an experiment was
performed on some fairly large crystals of graphite. These crystals
consisted of laminated fragments embedded in a calcareous material
which could be broken with comparative ease into small pieces. After
some difficulty a satisfactory specimen of graphite was secured. The
laminations of this specimen were parallel to its length, were very
numerous and closely pressed together, and appeared to be very free
from impurities. It was cut into the form of a bar, the ends copper
plated, and copper terminals were soldered to these ends. The dimen-
sions of the specimen were roughly 0.3 X 0.4 X 5.0 mm., and its resis-
tance was 0.08 ohm. The experimental results obtained with this sped-

VOL.X.1

v^]

RESISTANCE AND MAGNETIZATION,

373

Ob-

men are plotted in Fig. 4. Here the symbols ± and || mean that H is
respectively perpendicular and parallel to the normal to the laminae of
the specimen. These curves are similar to those of Figs. 2 and 3 except
for the relative magnitudes of the effects. Where a single crystal or a
single group of crystals is used it is evident that the magnitude of the
resistance change is greatly increased. In short, we may consider that
the experiments with the large crystal group corroborate conclusions
(i) and (2) above. Regarding con-
clusion (3) it is to be observed that
of the two lowest curves of Fig. 3 that
for the transverse field is the higher.
Another sample of crystalline gra-
phite tested out very carefully in this
respect gave similar results to those
plotted in Fig. 3. If we make the as-
sumption that the crystal is magnetic-
ally isotropic for all directions in the
plane of the laminations then we may
assert that a transverse magnetic field
produces a greater effect than a longi-
tudinal field when the structural changes produced by H are the same
in both cases.

The general conclusions, however, which are to be drawn from the
above study of a group of small crystals indicate very clearly that we
must inquire carefully into the crystalline nature of metals before at-
tempting to interpret experiments which are made on metals. The
point of immediate interest is to decide whether or not a wire of given
material is magnetically isotropic as regards directions parallel and
perpendicular to its length. Ordinary magnetic measurements are dif-
ficult to make with sufficient accuracy in this case, and furthermore, we
cannot be sure that isotropy as regards magnetic permeability, for
example, will guarantee isotropy as regards resistance change in a
magnetic field. A way in which we may gain information, however, is
as follows. If a thin section of metal is cut from a cylindrical bar of
cast metal (the cut being perpendicular to the length of the bar) and
then hammered out into a thin plate we may suppose from principles of
symmetry that for all directions in the plane of the plate we will have
magnetic isotropy.

A thin rectangular plate of cadmium was prepared in this way and a
series of parallel cuts made across it. These cuts did not completely
traverse the plate, but were so made that four very thin strips of metal

374 c. w. HEAPS, [ISS^

were secured, each about two centimeters long and half a millimeter wide,
lying side by side and joined in series. When experiments were made on
this specimen it was found that a transverse magnetic field of 7,550
gausses produced the same increase of resistance whether parallel or per-
pendicular to the surface of the strips. A difference in dR/R as small as
3 X io~* could have been detected in the two cases. Keeping the
magnetic field in the plane of the strips but rotating it so as to compare
the longitudinal and transverse effects, it was found that the transverse
field produced the greater increase of resistance. The average of two
trials gave

dR BR

_-_ = 9Xio-

with H = 7,550. This result agrees as well as could be expected with
previous experiments* made on a different sample of cadmium in the
form of wire. We may conclude then, that the cadmium strips were
magnetically isotropic with respect to resistance change. If strips of
metal hammered out in this fashion are magnetically isotropic it is prob-
ably safe to assume that cadmium wires are similarly isotropic as regards
resistance change.

It was difficult to test other non-ferromagnetic metals than cadmium
in this way for magnetic isotropy because those metals which are easily
worked with mechanically have a small resistance change in a magnetic
field, and hence require a more sensitive apparatus than the one available.
In a previous paper* the writer has measured both the transverse and
longitudinal effects in wires made of different metals. In all cases the
transverse effect was greater than the longitudinal as we should expect
from equation (9). If we assume that the other metals are like cadmium
in being magnetically isotropic we may apply equations (8) and (9) to
these substances and calculate the mean free period of the electrons and
the number of electrons per unit volume of metal. The following table
gives the results of the calculations, data being taken from the previous
paper except in the case of bismuth and graphite. In the calculations it
is assumed that e = 1.6 X io~^ e.m.u., e/m = 1.7 X 10^, and H = 8,000
gausses.

Patterson' obtained values of n somewhat different from those given
above but in his equation no account was taken of the longitudinal effect
or of the effect of H on molecular structure. Other methods of deter-
mining n are based upon Drude's theory of the optical properties of

» PhU. Mag.. Dec., p. 900, 191 1.

* Phil. Mag., Dec, p. 900, 191 1; Dec, p. 813, 191 2,

• Phil. Mag., 3, p. 643, 190a.

Vol. X.1
No. 4. J

RESISTANCE AND MAGNETIZATION,

375

dR SR

ff(e.m.u.).

R R

Te<*)

«)4.7 XlO-»

4.9 X10-»

S.1X10-"

5.0 X10»

Bi<»)

tt)0.84

0.144

39.2

11.8 XIO^'

PbS<«

«>0.042

1.15XlO-<

0.8

2.9 X10»»

tt)13.0

6.5 XlO-»

0.59

1.2 X10«

0)16.0

2.9 XlO-»

0.4

2.2 X10«

0)41.0

0.3 X10-*

0.12

18.8 X10«

Graphite <^

0)0.036

0.024

11.4

1.74X101'

* From Kaye and Laby.

* Baedeker, " Elektrischen Erscbeinungungen in Metallischen Leitem.

* Cast in cylindrical form.

» The tesU

on bismuth were made using a Hartmann and Braun spiral. The magnetic

field used in this case was 5.700 gausses.

* Natural crystal. Tests showed it to be magnetically isotropic.
' Large natural crystals of Fig. 4.

metals and upon the theory of thermionic emission. Experiments made
by Spence* on the refractive index of metals lead him to give as probable
values of n for gold the number 3.3 X 10**, for silver 3.7 X 10**, and for
platinum a number less than 8.1 X 10**. From measurements on the
thermionic emission of hot metals H. A. Wilson* deduces for platinum
a value of n = 1.5 X 10^. These results are of the same order of mag-
nitude as those tabulated above.

It is of interest to consider how equations (3) and (6) may be applied
to the experiments. In the case of all the substances listed in the above
table we must consider the term 5X/Xo to be very important, for in all
these materials the change of resistance in a longitudinal mag^netic field
is comparable with the change in a transverse field. We are thus led to
believe that in equation (6) the term dX/Xo must account for a large part
of the resistance change. Furthermore 6\ must represent a decrease
in free path, otherwise a longitudinal magnetic field would decrease the
resistance of the conductor — a, result contrary to experiment. It is to
be hoped that a more complete understanding of this term, 6X, will be
reached by further experiment.

Conclusions.

I. The theory of the effect of magnetic fields on resistance has been
developed to cover the cases of longitudinal and transverse magnetic
fields. The theory leads us to expect a greater transverse than longitu-
dinal effect. All experiments on non-ferromagnetic metals appear to be
in agreement with this conclusion. In order to make definite tests,

» Phys. Rev., a8, p. 337, 1909.

« "Electrical Properties of Flames."

iy6 C. W. HEAPS.

however, we are obliged to consider carefully the c
the metals examined.

2. With this fact in mind experiments have bee
graphite in different forms. In compressed grapl
of a magnetic field on the resistance is greater in •
fine. This fact implies that in a substance comp
the effect is greater than when it is composed of
probable that this effect of crystalline structure c
which in ordinary form suffers a large increase of n
field, is very little affected when in the form of fin
chemical reduction of a bismuth salt.

There is a greater effect in large crystals of gr
been discovered in any substance which is metallica
the study of graphite it is to be concluded that ■
important factor to consider when experimenting
any substance in a magnetic field. By making up s
way it is possible to compare the longitudinal
experimentally without any trouble arising from d
structure.

3. From experiments on a specially prepared s£
to be concluded that comparisons of the longit
effects may be made legitimately using an ordinary

Assuming that wires of other metals are like ca- -*-.* m cms respect
calculations have been made for several metals of the number of electrons
per unit volume and of the free period of the electrons.

4. The theory affords a satisfactory explanation of the behavior of all
para- and diamagnetic substances which have so far been examined.

The writer is indebted to Mr. N. S. Diamant, of the department of

engineering, for aid in securing graphite samples; also to the Dixon

Crucible Co. for kindly supplying crystals of graphite in a natural

condition.

The Rice Institute,
Houston, Texas.

V«L.^.J TRUE TEMPERATURE SCALE OF TUNGSTEN, 377

THE TRUE TEMPERATURE SCALE OF TUNGSTEN AND ITS
EMISSIVE POWERS AT INCANDESCENT

TEMPERATURES.

By a. G. Worthinc.

Introduction.

A TRUE temperature scale for tungsten at incandescent temperatures
based on sound principles was first obtained by Pirani^ when he
bent a tungsten ribbon back and forth so as to obtain a cavity largely
surrounded by it, which was raised to incandescence in the ordinary
manner by a heating current. He concluded that the radiation coming
from the interior of the cavity was black body radiation. Thus he was
able to express, with the aid of a Holborn-Kurlbaum optical pyrometer,
the relation between the brightness temperature* of the natural tungsten
radiation and the true temperature of the tungsten, that is the brightness
temperature of the radiation from the cavity. An emissive power
relation follows simply. He concluded that for X = 0.64M the emissive
power was constant and equal to about 0.485. However the uncertainty
as to this result was rather large, being stated as 7^ per cent.

Soon afterwards Mendenhall and Forsythe' used a narrow V-shaped
trough and in a similar manner obtained a temperature scale which in-
volved emissive powers increasing with temperature from 0.45 at 1 100® C.
to 0.66 at 2900** C. Unfortunately their results were subject to consider-
able error because the two strips separated at the apex of the V at high
temperatures.

Two other scales based on somewhat similar pyrometer measurements
have been developed by Pirani and Meyer* and by Langmuir.* In both
instances the brightness of the interior of a closely wound tungsten helix
was compared with the brightness of the exterior. Pirani and Meyer's
results indicate that the emissive power at the wave-length 0.532^11, which

1 Phys. Zeit., 13. p. 753. 1912.

* Heretofore the term "black body temperature" has been used to designate this quantity.
The reasons for abandoning this term in favor of "brightness temperature" are fully stated
by Hyde, Cady and Forsythe in the paper following this.

* Astrophys. Jour., 37, p. 380, ipis-

* Elektrot. u. Masch., 33, pp. 397 and 414. 1915.

* Phys. Rbv.. II., 6, p. 138, 1915.

378 A, G. WORTHING. [g^S

seems to have been used by them, is constant with temperature and
equal to 0.44. Langmuir concluded that the emissive powers at 0.667/x
and 0.537M were independent of the temperature and equal respectively
to 0465 and 0.485. Later^ he concluded 0.46 to be the most probable
value for the wave-length 0.664/x. The difference between the two
scales for 0.532)11 and 0.537^ is considerable. The earlier temperature
scales which these experimenters had arrived at, and to some extent,
the later scales just reported, have been founded on or tempered by the
results of Holborn and Henning* who concluded that the emissive powers
of silver, gold, platinum, and palladium in the visible spectrum were
independent of the temperature.

Shackelford' working in this laboratory, using helical coils of varying
pitch, and extrapolating for the case of a closed helix obtained, for a red
brightness temperature of 2300° K. (true temperature of 2530° K.),
0.445 2ts the emissive power at 0.656^ and 0.465 at 0.493^. At a tem-
perature about 400° lower values slightly larger were obtained. Others
have measured the emissive power of tungsten at some one temperature.
These are well summarized by Burgess and Waltenberg* who obtained
0.39 at 2020° K. for 0.65M. Considering further only the later values
as the more probable, there are in addition Coblentz's* 0.474 at o.65/i»
Wartenburg's* 0.51 at 0.65M and Littleton's^ 0.545 at 0.589/1, all of which
refer to room temperature. Other temperature scales depending on the
fact that the luminous flux from a tungsten filament may be matched in
color with that from a black body are discussed in the following paper
by Hyde, Cady and Forsythe.

Method and Apparatus.

General Procedure, — In the present work, except for the measurements
at room temperature, as will appear later, long tubular filament of tung-
sten with small holes penetrating the side walls at various places have
been made use of. In general terms, the procedure has consisted of
determining with an optical pyrometer the ratio of the brightness of the
filament surface adjacent to a hole, in a region suitably chosen from the
standpoint of constancy of temperature, to the brightness of the hole,
when the filament was heated to incandescence in a vacuum or in an

» Phys. Rev.. II.. 7, p. 302, 1916.

* Berl. Ber., p. 311, 1905.

'Jour. Frank. Inst.. 180. p. 619. 1915. and Phys. Rev., II.. 8, p. 470. 1916.

* Bull. Bur. of Stds.. ii. p. 591. 1915.

* Bull. Bur. of Stds.. 7, p. 197. 191 1.

* Verh. der Deut. Phys. Gesell.. 12, p. 105. 1910.
7 Phys. Rev., 35, p. 306, 191a.

Na*4^'] TRUE TEMPERATURE SCALE OF TUNGSTEN, 379

atmosphere chemically inert. On the assumption that the radiation
from the hole is black and that there is a negligibly small difference of
temperature between the interior and the surface, such a ratio represents
an emissive power for a wave-length depending on the light transmitted
by the pyrometer glass screen, and for a temperature corresponding to
that of the radiation from the hole. This latter temperature was ob-
tained in the standard manner with the aid of Wien's law by comparing
the black radiation with that from a calibrated black body of the ordinary
type at the palladium point. As already noted a brightness temperature
true temperature relation follows simply. The assumptions made and
the corrections for the lack of their fulfillment are considered in detail
in the section on Corrections and Errors.

Preparation of Filaments. — ^The filaments themselves were formed
according to the commercial method common about five years ago by
squirting a paste of tungsten powder held together by a binder through
a die, in the present case one with an annular opening. Shortly after
the actual squirting, when the tubes were of the proper consistency,
small holes were pierced in many places through the walls. Following
the usual drying and heating, the filaments were mounted in lamp bulbs.
At this stage the filaments used had external and internal diameters of
about 1.3 mm. and 0.8 mm. respectively. The holes through the walls
were nearly circular and of two sizes, approximately 0.09 mm. and 0.12
mm. in diameter.

Much difficulty has been experienced in mounting these filaments in
lamp bulbs because of the extremely large currents, 100 amperes being
the maximum, which were required. It was considered impracticable,
after several failures, to use soft glass bulbs. Many successful lamps
using hard glass have been made, but the bulbs of those made at first
contained so many bad streaks that it was often impossible even with
selected bulbs, to obtain observations on more than one or two portions
of the filaments. The later lamps have been fairly satisfactory in this
respect, however. Some of the bulbs have been evacuated, but most
of the data reported have been on filaments immersed in a gas, usually
argon.

Apparatus. — ^The optical pyrometer used was of the Holborn-Kurlbaum
or Morse type such as has been used quite commonly in this laboratory.^
As pyrometer screens a red glass, Jena F-4512, and a blue uviol glass in
single and double thicknesses have been used. As will be shown later,
the changes in thickness and the lack of monochromatism were readily
corrected for, so that the final results may be considered as applying

^ Pbys. Rbv., II, 4, p. 163. 1914.

380 A. G, WORTHING, [iSSS

Strictly to two definite wave-lengths, viz., 0.665/4 and 0.467/*. Photo-
graphs are reproduced in Fig. i, which show considerably magnified
what is seen when the pyrometer filament appears somewhat less bright
than the hole but brighter than the adjacent surface for each of the
two sizes of holes used.

Preliminary Tests.

As is usually the case, these tests are preliminary from the standpoint
of character rather than from the standpoint of time of performance.
They represent tests which were essential, before any reliability could be
placed upon the main experimental data obtained.

Uncertainty Due to Smallness of Holes, — Fig. i suggests that, due to the
smallness of the holes, from physiological and psychological grounds,
one might be expected to make erroneous judgments, thus vitiating the
results. In order to test this, pyrometer settings were made on an
extended luminous background such as was described by Lorenz,^ first
when viewed through a fine needle hole in an opaque screen just in front
of the background, then when viewed without the opaque screen. With
the fine needle hole of approximately the same size as the holes in the
filaments used, no systematic differences in the pyrometer readings, which
depended upon the presence or absence of the opaque screens, were notice-
able. The same conclusion as to freedom from error on this account was
borne out by the results obtained with changes in the magnification under
workable conditions.

Distortion in Temperature Distribution Due to Presence of Holes, — Un-
questionably the presence of such holes as were pierced in the walls of
the filament caused variations in the temperature distribution in their
neighborhood. Many times tests for such changes including settings
as close as 0.02 mm. to the edge of the hole were made, but in no case
was such an effect detectable.

Constancy of Temperature of the Surface on a Given Circumference, —
Broken filaments showed in general that the inner and the outer surfaces
of the filament wall were not coaxial, but that the maximum and mini-
mum thicknesses of wall at any one cross-section varied as much as 7
to 5. Attempts to determine the effect of this mathematically have
been unsuccessful. Pyrometer settings at various positions around a
circumference revealed no certain differences at temperatures above 1500®
K. Data at lower temperatures were not conclusive. It is further
believed that in the average any effect of this type would be eliminated.

Effects of Preliminary Heating, — The first results obtained showed that

> Elec. World. 61. p. 932. ipiS-

PHViiicAL Rbview, Vol. X., Second Series, Plate 1.

October, 1917. To face page 380

Fig. 1.
PhotoErapli« slmwiiig thi- iij-nimi'trr lllnmrnt ptiijcclcrl against the hole ai)<] the mirface a:
J, liacliBtound tor each of thu two siies of holes used.

■ A. O. WORTHING.

No!"^l TRUE TEMPERATURE SCALE OP TUNGSTEN. 38 1

a gradual change was taking place. With continued operation the
values obtained for the emissive powers gradually decreased, the total
change amounting to as much as 7 or 8 per cent, of the quantity measured.
It was soon found that the filament could quickly be brought fairly close
to its final steady state by a preliminary heating for a short time at a
high temperature. In subsequent work such preliminiary heat treatment
at about 2800® K. was always given, usually previous to the completion
of the lamp, while it was still connected with the evacuating pump.
That such a temperature was reached was assured by a method cbmmon
in this laboratory, in which there were compared the colors of the two
shadows of a pencil or some slender opaque object on a piece of white
paper as produced by the lamp being tested and an ordinary commercial
loo-watt gas-filled tungsten lamp. It is necessary to have the two
shadows about equally bright. In the heat treatment given, the tem-
peratures reached were always such as to indicate that the color of the
light from the lamp tested was noticeably bluer than that from the com-
mercial lamp.

Effects of Lack of Surface Polish, — The accidental short-circuiting of a
resistance, which caused the filament being studied to melt at a certain
cross-section, was apparently also the means whereby the surface was
partially polished. The subsequent tests with this filament seemed to
give lower values for the emissive power than were obtained previously.
Later tests on polished and ordinary unpolished filaments showed this
effect to be real and to account for differences which may amount to
two per cent, in the emissive power. By variations in the process of
preparing squirted tungsten filaments, filaments having various surface
appearances may be obtained. For definiteness of results the need of
specifying the surface character cannot be overstated. In illustration
of this it is sufficient to say that the writer has in his possession filaments,
the structure of which is such that the surface has a large diffuse reflec-
tivity. Emissive powers for these filaments as ordinarily measured are
of the order of 50 per cent, greater than those for polished filaments.
Because of these considerations, the determinations of emissive power
to be reported have been confined to polished or fairly well polished
material.

Effective Wave-lengths of Blue Glass Screens, — ^A further preliminary test
consisted of determining the effective wave-lengths of the blue uviol glass.
The red glass had previously been studied by Hyde, Cady and Forsythe.^
As defined by them, the effective wave-length for a screen for a definite
temperature change in the source viewed is that wave-length for which

1 Astrophys. Jour.. 42, p. 294, 191 5.

382 A. G. WORTHING. [i

the relative change in monochromatic brightness is equal to the relative
change in total brightness for the luminous flux transmitted by the
screen. Following a method much as reported previously^ and as more
fully outlined by Langmuir* the effective wave-lengths of a single and of
a double thickness of the blue glass have been determined for different
temperature intervals of black body radiation. By platting the logs of
the blue brightnesses as a function of the logs of the red brightnesses for
ranges of 100 to i of the latter for black body radiation, a very good
straight line relation was found. The slopes, using one or two thicknesses
of the blue glass against two of the red glass, were respectively 0.745 ^^^
0.706 indicating that if for black radiation for a given temperature inter-
val, the effective wave-length of a screen composed of the two red glasses
is, say, 0.665/*, the corresponding effective wave-lengths for the blue glass
are respectively 0.495/* ^^d 0.469/*. Unfortunately, the method is in-
sensitive in showing variations in effective wave-lengths. All that may
be said is that the effective wave-lengths thus determined are average
values. The variations with temperature, for the blue glass using the
method described by Hyde, Cady and Forsythe, were found to be fully
five times as great relatively as those they found for the red glass.
Further considerations due to the lack of monochromatism in the trans-
mission of the screens will be considered in the next subdivision.

Corrections and Errors.

Difference in Temperature between the Inner and Outer Surfaces, — ^A
simple formula given in a paper by Angell' expresses this difference in
terms of the thermal conductivity and of ordinary measurable quan-
tities. Letting

fo = external radius of the hollow filament,

r» = internal radius of the hollow filament,

i = current density,

p = resistivity,

k = thermal conductivity,

T = temperature,
AT = increase in temperature in passing from the external to the internal
surface,

E = radiation intensity,

B = brightness,
Bx = brightness ordinate at X,

* Astrophys. Jour., 36, p. 348, 1912.

* Phys. Rev., II., 6, p. 146. 1915.
» Phys. Rev., II.. 4, p. 535, 1914.

VOL.X.

^•] TRUE TEMPERATURE SCALE OP TUNGSTEN, 383

Ct = constant in Wien's equation,
€0 = observed emissive power,
€ = corrected emissive power,

subscript (v) refer to filament in vacuo,
subscript (ar) refer to filament in argon,
we have

2k\ fi 2 I

For the filaments used for the greater part of the work r© and f» were
respectively 0.66 mm. and 0.38 nun. Taking account of the simply
derived relation

we have

E = vp,

2ro

AT = T X (0.040 cm.).

The effect of this AT" on observed emissive powers is seen when one com-
putes with the aid of Wien's equation, the relative increased brightness
of the hole resulting from the existence of this AT". Thus

B^ dT xr«

Evidently also

To obtain values applicable to the gas-immersed filament, it is only
necessary to multiply the correction here found by the ratio of the square
of the current when thus inmiersed to the corresponding value for the
filament in a vacuum. Values of k and E for tungsten taken from a table
appearing in a later subdivision lead to the results given in Table I. The
effects of these corrections will be shown later.

Lack of Blackness in Radiation from the Hole. — ^There are three factors
tending toward departure from perfect blackness in this radiation, (i)
the presence of the small hole for observing, (2) the existence of a tem-
perature gradient along the tube, and (3) the presence of possible crystal
surfaces on the inner surface of the tube.

In connection with the first factor, it is easy to compute the departure
from blackness on the supposition of a long tube of uniform temperature
with a perfectly matt interior surface. For the smaller of the two sizes
of holes specified, and with an assumed emissive power of 0.45, it follows
that the radiation will deviate from blackness quantitatively by about

384

A. G. WORTHING,

rSlCOHD

Table I.

Emissive- power Corrections for ike Temperature Difference between ike Internal and External

Surfaces of the Filaments Used,

AT,.

Aso.665fi.

A -0^67^.

T 9Bk
Bk9T'

14.4

9.4
7.0

V «0 /•*

T 9Bx
BKdT '

m.-

1500'' K.

2300

3100

0.22*'

1.54

5.6

.002
.006
.012

.004
.007
.013

20.5
13.4

9.9

.003
.009
.018

.006
.010
.019

O.I per cent.; for the larger size holes it will be about two times this or
0.2 per cent.

Quantitative computations regarding the effects of the second factor in
producing a departure from blackness are difficult. Measurements have
• been made almost entirely on portions of the filament where the temper-
ature was constant to within a few degrees over lengths on each side for dis-
tances of at least five times the internal diameter of the tube. Moreover,
measurements intentionally taken where a noticeable temperature
gradient existed did not yield results noticeably different. Errors from
this source will be more noticeable at the low temperatures than at the
high temperatures, because the cooling effects of the supports and leading-
in wires are confined to shorter lengths of the filament at the higher
temperatures, these lengths being inversely proportional to the heating
currents. Errors from this source are probably very small.

The third factor tending away from blackness was a matter of some
concern in connection with a certain filament, particularly following
the short-circuiting of a resistance in series with it and the consequent
melting of a portion of the filament as previously mentioned. Dark
irregular patches were noticed within the holes. Later microscopic
inspection of the filament showed the surface to be made up of com-
paratively large crystal surfaces. The accidental orientation of such
crystals normal to the line of sight on the inner wall and in line with a
hole were apparently the explanation of the dark patches mentioned.
The occurrence of a large number of such surfaces oriented irregularly
is of course equivalent to a matt surface, such as has been considered
already. In the experimental work, by arbitrarily orienting the filament,
the spots were eliminated from the field of view. Results with this
filament were not noticeably different from those with other polished
filaments.

Except in giving the lower values for the emissive power at a given
temperature greater weight than the higher values, no correction for
these departures have been made.

VOL.X.

Na4>

]

TRUE TEMPERATURE SCALE OP TUNGSTEN.

385

Lack of Monochromatism in the Light Used. — It is necessary to consider
here to what wave-lengths to ascribe^ the brightness temperature meas-
urements made with the roughly monochromatic screens used in optical
pyrometry, and how to correct the emissive power determinations made
so that they will uniformly apply to a single wave-length.

Consider in this connection Fig. 2, in which in an exaggerated way
curves a, /3, 7 and S represent for a given filament at a temperature T,
certain spectral brightness B^, distributions related to the luminous
flux transmitted through the pyrometer system including the colored-
glass screen at the eyepiece. Let a refer to the black body radiation at

Fig. 2.

A diagram showing various spectral brightness distributions connected with tungsten
filaments such as used, which are helpful in determining the wave-length to which to ascribe
brightness temperature measurements.

the temperature T coming from a hole in the filament wall; /3 the natural
tungsten radiation arising from the adjacent external surface; 7 the
radiation from a black body having the temperature 5, the measured
brightness temperature of the natural tungsten radiation; and S that
black body radiation whose relative brightness distribution is the same
as that given by p. These diagrammatic distributions assume the pos-

1 The writer is in part indebted to his colleague W. £. Forsythe for this development.

386 A,G, WORTHING, [^S

sibility of color matching the tungsten radiation with black body radi-
ation. Thus curve h is, according to Hyde, Cady and Forsythe,^ the
brightness distribution of a black body at a temperature given by the
color temperature of the natural radiation. Evidently from the definition
of brightness temperature the areas included under curves fi and y are
equal. It is also evident that only at the wave-length X' is the brightness
temperature of the natural radiation equal to 5, being progressively less
than S as the wave-length is increased beyond X' and progressively
greater than 5 as the wave-length is decreased below X'.

Representing by ^x* fi^K* etc., values of -B^ corresponding to curves

I .-BxdX, etc., we then

have

i^x «-B ^B ^By*

where in the first member X, of course, refers to any wave-length within
the range concerned. The last of the above equations according to
Hyde, Cady and Forsythe* is also the defining equation of the effective
wave-length of the pyrometer screen for black body radiation for the
temperature interval given by curves y and S. It follows therefore that
the wave-length X' to which the brightness temperature 5 is to be ascribed
is the effective wave-length of the screen for black radiation in going from
the brightness temperature of the tungsten to its color temperature. In
the writer's work X' for tungsten has varied from 0.6662/* at i6oo** K.
true temperature to 0.6628/* for 3200° K.

Having once determined X', the method of determining 5o, the bright-
ness temperature which shall correspond to some common wave-length
Xo arbitrarily chosen, is simple. It consists in finding the temperature of
a black body corresponding to 70 (Fig. 2). 70 must evidently intersect
/3 at Xo. The application of Wien's law to a change in which

ff°Ao _ Yo°Ao __ <°Ao

pBx' yBf^f iBx'

gives the result desired. Choosing Xo as 0.665/* means in the writer's
work that the values of 5o — 5 for tungsten for the red light are re-
spectively + 0.2° and — 1.4° at true temperatures 1600** K. and 3200° K.
The corrections for the blue uviol screen are somewhat greater.

In a similar way the wave-length to which to ascribe the emissive

* See following paper.

* Astrophys. Jour., 42, p. 294, 1915.

Na'4^1 TRUE TEMPERATURE SCALE OP TUNGSTEN. 387

power measurements, may be determined. Imagine another spectral
brightness distribution curve added to the somewhat complicated figure,
which shall enclose underneath it an area equal to that enclosed by p,
and which shall bear the same relation to a that p does to S. Call this
curve P'. The ratio of its ordinates to that of a will everywhere be equal
to the measured emissive power. It will cross the curve P at some wave-
length X". Evidently at this wave-length only is the ratio of the ordi-
nate of P to that of a equal to the measured emissive power. Hence
strictly the emissive power measured should be ascribed to X". As in
the case of X' just described, X" may be shown to be the effective wave-
length for the optical system in passing from distribution a to distribution
8. X" is slightly shorter than X'. On considering later the change in
emissive power in going from 0.665X to 0.46 7X together with color match-
ing possibilities, it will be seen that the changes in the emissive power
in going from X" to Xo are very small. In this work such corrections at
0.665/4 were inappreciable, those at 0.467/* were just appreciable, as will
appear later.

At temperatures below 1500° K. in the case of the red light and below
1700° K. in the case of the blue light, single thicknesses of pyrometer
screens were used. The corrections to be applied according to the fore-
going principles in order to obtain values to be expected if the regular
double thicknesses has been usable, are appreciable but not large.

Standardizations.

Any expression of emissive power as a function of temperature neces-
sarily implies a temperature scale which in turn is based on certain stan-
dardization points. In the preliminary notice of this paper^ the tem-
perature scale was based on 1336° K. and 1822° K. as the melting points
of gold and palladium respectively. As shown by Hyde, Cady and For-
sythe,* this with the assumption of Wien's law, which in its effects is
indistinguishable in the visible spectrum from Planck's law, leads to a
C2 of 14460/* X deg. For reasons stated elsewhere' our laboratory has
abandoned this scale and adopted that one based on 1336** K. as the
melting point of gold and 14350/* X deg. as the value of d. This leads
to 1828° K. as the melting point of palladium. The importance of
stating these underlying bases of temperature scales when one is men-
tioned should be strongly emphasized. The temperature measurements
in the present work were carried out with the aid of a large tungsten

* Jour, of Franklin Inst., 181, p. 417, 1916; Phys. Rev., II., 7, p. 497, 1916.

* Astrophys. Jour., 42, p. 300, 1915.
« Gen. Elec. Rev. — to appear soon.

388 A.G, WORTHING, [ISSS

filament lamp, which had been standardized as to brightness temperature
at the palladium point by W. E. Forsythe, of this laboratory. The
calibrations of the sectored disks used in conjunction with Wien's law
in determining other brightness temperatures through comparisons with
the standardized palladium point were made by a photometric method
by F. E. Cady, also of this laboratory and are believed to be known in
consequence of repeated determinations and checks with an accuracy of
the order of o.i per cent.

The method of determining temperatures is given by the following
equation

where X = the effective wave-length,

5o = the brightness temperature of the standard (palladium point),
5 = the brightness temperature being determined,
Bx = the brightness ordinate at X of the spectral brightness distri-
bution curve,
oBx = value of Ba corresponding to 5o,
Ci = constant in Wien's equation,
/ = transmission of the sectored disk used.

Results.

In the present paper only emissive powers in a direction normal to the
surface or nearly so are considered. Values for other angles of emission
may be computed with the aid of measured values of the deviation of the
radiation from Lambert's cosine law.^ The values there referred to,
however, were obtained on unpolished material and must be so considered.

The experimental values obtained, except that those obtained with
single thicknesses of pyrometer glass have been corrected as described so
as to refer to double thicknesses, are platted in Fig. 3. Points indicated
by difference symbols represent values, as per the accompanying caption,
obtained with different filaments or possibly the same filament with the
surface renewed by polishing. At room temperature a different procedure
was followed. Here for the most part a polished filament previously used
by Weniger and Pfund in infra-red measurements and discarded because
of pits formed in use, and to some extent some mirror surfaces formed
by melting carefully the larger portion of the ends of tungsten terminals
in an arc lamp* were used. Both types of surfaces had previously the
preliminary heat treatment already mentioned. The reflectivity was

1 Astrophys. Jour.. 36. p. 345, 1912.

"Langmuir. II., 6. p. 138, 1915; Luckey, Phys. Rev.. II., 9, p. 132. 1917.

Vot.X.1
Na4. J

TRUE TEMPERATURE SCALE OP TUNGSTEN.

389

measured in the ordinary way, using the pyrometer apparatus as in the
previous measurements. The measurements consisted of brightness
determinations of a definite spot on a broad lamp filament, first when an
mage of the filament was viewed directly, then when viewed reflected
from the polished surface, there being, of course, identical optical paths

7>^

Fig. 3.

Emissive power results for tungsten as a function of the temperature at 0.665M and o.467m«
X. values obtained on unpolished filament in much striated bulbs.
+, " " " polished filaments in much striated bulbs.

O, " " " polished filament in fairly clear bulbs.

•, " " at room temperature by reflection method,

a, a\ weighted curves for data obtained.

y, curve a' corrected for lack of monochromatism of the uviol glass.

c, c', final curves containing corrections for differences in temperature between interior and
exterior surfaces of the filaments.

in the two cases, except for the reflection from the polished surface. It
was surprising to note how much the image of the broad lamp filament
formed at the surface of the discarded polished filament was broken up
by fissures, and at the same time that it was impossible to see any such
fissures at all when the filament was self-luminous and viewed normally
or nearly so. This indicates that for normal emission generally there

390 A, G. WORTHING, [

was no opportunity for blackening of the radiation. A consequence of
this is that a rather rough polish of surface only is necessary for emission
measurements normally on a self-luminous filament, a fact quite in con-
trast with the requirements for reflection measurements.

The earlier measurements were made on unpolished filaments in lamp
bulbs which, as has been stated, distorted the images somewhat in almost
every instance and did not always permit of the selection of holes entirely
satisfactorily located from the standpoint of end cooling effects. In the
later measurements a partially polished filament was used and the bulb
was such as to permit of undistorted images. For these reasons in draw-
ing the curves much emphasis has been given to the later measurements.
Further, because errors due to lack of blackness in the radiation from
the hole tend toward too high values of the emissive power, the lower
values have been given greater weight than the higher values. The
heavy lines a and a' in Fig. 3 show the weighted results. Line ft' repre-
sents the curve obtained when corrections are made for lack of mono-
chromatism in the pyrometer screen transmitting blue light. As stated
previously in the case of the red light this correction is negligible. Curves
c and c' represent the final emissive power curves in which corrections
have been made for the difference in temperature between the interior
and the surface of the filament.

When once an expression between temperature and emissive power
for a substance is obtained, the use of Wien's equation enables one to
express directly the true temperature as a function of the brightness
temperature, i. c, the temperature scale for the substance. Thus

The relation between T and S for tungsten is given in Table II. by
steps of 200°. The highest temperature refers to the melting point of
tungsten under atmospheric pressure, further considerations concerning
which appear below. The last column of the table indicates the uncer-
tainty in the true temperature of tungsten to be ascribed to an uncer-
tainty of I per cent, in the emissive power — that which is considered as
probable for the results here presented — ^when computing the true
temperature from brightness temperature and emissive power measure-
ments. It gives at once a method of comparing the scale here obtained
with that of others. It is readily seen that Langmuir's scale, when shifted
so as to agree as to fundamental characteristics, that is as to the gold
point temperature and Cj, and which is based on a constant emissive
power of 0.46 agrees with the writer's at iioo® K., but differs from it

VOL.X.

Na4*

]

TRUE TEMPERATURE SCALE OP TUNGSTEN.

391

Table II.

Temperature Relations for Tungsten on Basis of d - i43S0fi X deg, and Tau - 133^^ K,

St
Anso.665fi.

«mt

Asso.66sM>

TS.

AT* in Cam

^=+0.01.

1200

.457

-0.7

1400

.451

1.0

1600

.446

102

1.3

1800

.440

132

1.7

2000

.434

168

2.2

2200

.428

208

2.7

2400

.422

254

3.3

2600

.416

306

3.9

2800

.410

^ 366

4.6

3000

.403

433

5.4

3176 (melting point)

.398

498

6.2

at 2400** K. (the approximate operating temperature of an ordinary
40-watt vacuum tungsten lamp) and at 3675° K. by 18® and 88° re-
spectively. No similar comparison can readily be made with Pirani and
Meyer's scale since they used a very different wave-length, but their
result of a constant emissive power of 0.44 at 0.532/4 is seen from Fig. 3
to be consistent with the writer's only at a temperature in the neighbor-
hood of 2400° K. However, data obtained by Schackelford^ on emissive
powers in the visible region with the aid of helical filaments of various
pitches, by Hulbert* both as to changes in emissive power with wave-
length and with temperature in the ultra-violet region, and by Weniger
and Pfund* on the reflecting power of tungsten are in very good agree-
ment with those here presented. It is a point worth emphasizing that
the tungsten used by them was in the form of wire which had been
drawn as in the common commercial method of preparing tungsten
filaments while the writer used the squirted paste filaments. Of the
remaining individual emissive power values mentioned in the introduc-
tion, only that one given by Coblentz for room temperature, 0.474 at
0.65/1, 5s in good agreement with those presented here.

The Melting Point of Tungsten.

The brightness temperature of tungsten at the melting point as
recorded in Table II. represents the mean of the four results shown in
Table III. Other results on the melting point of tungsten have been
summarized by Langmuir and Luckey in their papers. Only the four

* Loc. cit.

* Jour. Frank. Inst.. 182, p. 695, 1916; Astrophys. Jour., 40. p. 149. 1917.

* Jour. Frank. Inat.. 183, p. 354. ipi?-

392 A, a WOSTBING,

results mentioned have been included, ancc in connection wirii these
only are the methocia aoond axid the knovrledge defimtF ats to effective
wave-Iengthg used and aa to the wave-length to which to ascribe the
reailta. The two methods of determining the brightneae temperature at
the melting point have bem weU described by Langmuir. The writK

Table III,

Data^ cm yidting^Pomt of Tmm^Un <m Basis of Cx ^ i4350iL X deg, mmd r«» » ijj^ X

Mcndcnhall&Foraythc*.. I 3174^ K. , Faameut mclta.

Langmuir* 3187 ! Filament meits and molten arc tsnninais.

Worthing* ,
Luckey*. .

Av. . . .

3174 I Molten arc terminals.

3169 Molten arc terminals^

3T76 !

^ Oata were obtained at Nela Research. Laboratory in aommer of 1914 but reaoita have
not been published heretofore.

»Pbys. Rxv., IL. 6. p. 153, 1915.

» Joor. Franidin Inst., 18 r, p. 417, 1916. Phts. Rxv., IL, 7, p. 497, 1916.

•Pars. Rhv., IL. 9, p. 133, 1917-

haa been informed by Langmuir that in his measurements on molten arc
terminals, the angle of emission varied considerably from the normal.
In consequence of the deviation from Lambert's cosine law, higher vahies
are to be expected than if the surface had been viewed normally. However,
both Lnckey and the writer in their determinations viewed the surfaces
normally or nearly so. This might in part explain the high valtse ob-
tained by langmuir on the molten arc terminaL However, there rpmains
as unexplained his still higher value from the filament melt data. Con-
siderations of effective wave-lengths brought forth in a subecqixnt paper
by Hyde, Cady and Forsythe^ together with certain considerations noted
above indicate that Langmuir's results on a basis of Ct = 14350^ X deg,
should give as an average 3191'' K. for 5 and that this is to be ascribed
to 0.66 iM' Reducing the results of all so as to refer to 0,665/4 has fed
to the results shown. An equally weighted average has been accepted
for the final result. Making use of the emissive power curve here pre-
sented, 3674° K. or in round numbers 3675° K. results as the true temper-
ature for the melting point. The uncertainty as to this, granting the
fundamental bases of the temperature scale, would seem to be not greater

than 15°.

Effect oy Previosly Published Reslxts.

The results on thermal and electrical conductivity and Thomson effect
previously obtained, expressed in terms of the new temperature scale,

^ Loc. cit.

VOL.X.1

Na4. J

TRUE TEMPERATURE SCALE OF TUNGSTEN.

393

are incorporated in Table IV. The radiation intensity values (see also
Fig. 4) in reality are the results of measurements at various times on five

Table IV.

Previous Data Corrected to New Temperature Scale.

Tin°K,

. . watts

*in 3 — .

cm.xdeg.

in CO.S. UniU.

. microvolts

ain — 3 .

degree

P in ^'"*

TdE
E df

1500

1.01

2.80^

5.7

5.21

1700

1.07

3.06

10.8

5.06

1900

1.12

3.29

-20

18.8

4.93

2100

1.17

3.50

-24

30.6

4.81

2300

1.21

3.69

-28

47.2

4.70

2500

1.25

3.87

69.7

4.60

2700

1.29

4.02

98.9

4.50

' It ia to be noted that the values originally published were in error by the factor 10.

filaments as indicated by the different symbols used in the plat. The
results in all cases are free from effects due to cool filament terminals.
Three of the lamps possessed very fine potential leads of tungsten wire
tied to the larger filaments. The remaining two lamps each possessed

^% /\

2.0

1.8

1.6

•

5 1-4

^^o

^ I.O

0.8

0.6

3.14 3.18 3.22 3.26 3.30 3.34 3.38 342

Log of T in ""K
Fig. 4.
Radiation intensity of tungsten as a function of temperature.

394 ^- ^- WORTHING, ^SS

two filaments with separate leads which differed only in length, so that
by taking differences end effects were eliminated here also. These
results may be expressed by the empirical equation

log E = 1.379 + 4.87(log T - 3.3) - i.4(log T - 3.3)^.

It has been assumed that, for practical purposes, the bulbs of the lamps
containing these filaments were at negligibly low temperatures. The
relative rate of change in emission intensity with relative change in
temperature is given under {T/E){dE/dT^ in the table. This quantity
for a black body is the exponent 4 occurring in the Stefan-Boltzman
equation. The results show a progressive approach toward the black-
body radiation in this one respect, but not much significance is to be
attached to this since just as fundamental a progressive deviation from
black body radiation is shown by the emissive power variation in the
visible spectrum.

Summary.

1. A method of determining the emissive power of a substance at
incandescent temperatures has been described.

2. A method has been described for determining the wave-lengths to
which brightness temperature and emissive power measurements made
with the aid of colored glass pyrometer screens are to be ascribed.

3. The emissive power of tungsten at 0.467/* and 0.665/* ^s a function
of temperature have been determined for temperatures up to 3200° K.
(Fig. 2 and Table II.).

4. The relation between the true temperature and the brightness
temperature at 0.665/i for tungsten has been computed (Table II.).

5. Determinations of the melting point of tungsten have been made.
From a consideration of these and other data, 3675° K. (Cj = 14350/*
X deg., Tau = 1336° K.) has been obtained as the most probable value
for this constant.

6. The radiation intensity as a function of the temperature has been
determined for tungsten (Table IV.).

7. Previous data on thermal conductivity and on Thomson effect
have been recomputed on the basis of the new temperature scale (Table
IV.).

Nela Research Laboratory,

National Lamp Works of General Electric Company.
Nela Park, Cleveland, Ohio.
June, 1917.

^^'] COLOR TEMPERATURE SCALES. 395

COLOR TEMPERATURE SCALES FOR TUNGSTEN AND

CARBON.

By E. p. Hyde. F. E. Cady and W. E. Forsythb.

L Introduction.

TN a paper published in 1909 by two of the present authors^ in collab-
-^ oration with Middlekauff, a new method was proposed and applied
for studying the selective radiating properties of certain metals such as
tungsten, tantalum and osmium as compared with a black body or with
untreated carbon which approximates a black body in its radiation. This
method was based on comparative measurements of the ratio of visible
to total radiation for the substance to be investigated and for the black
body when the two radiating bodies were heated to such arbitrary un-
known temperatures that the integral coloi* of the visible radiation was
as nearly as possible the same for the two. This ** color match " method
was further elaborated in subsequent papers,* and a more rigorous def-
inition of the " color match " criterion was given.

One of the earliest methods of estimating temperatures in industrial
work was based on rough eye observations of the color of the hot body,
but this has given place largely to other methods of greater precision
based upon different principles. In 1907 Morris, Stroude and Ellis,'
in a study of the relative operating temperatures of different incandescent
lamps, assumed as a starting point equality of temperatures when the color
of the light from the various lamps was the same, owing, as they state,
to '* the great divergence in the figures published by various experimenters
for this quantity." In the same year Leder* obtained the distribution
of energy in the emission of the Hefner lamp by determining the temper-
ature of the black body when it had the same energy distribution in the
visible spectrum. But in neither of these investigations was the color
match method employed as a means of studying the radiating properties
of metals, nor was any consideration given to the significance of the

color temperature " scale, or to its relation to true and black body

brightness temperatures " of incandescent metals.

1 Trans. Ilium. Eng. Soc., 4, p. 334. 1909. Presented before Am. Phys. Soc., Oct.. 1908.

* Jour, of Frank. Inst., i6q, p. 439, and 170, p. 26, 1910; Astrophys. Jour.. 36, p. 89, 191a.

» Elec., SQ. p. 584. 1907.

* Ann. d. Phys., Ser. 4. 24, p. 305, 1907.

396 £. p. HYDE. P, E, CADY AND W, E, FORSYTHE. [MSS2

It has been customary for some years past to give a number to indicate
the temperature of any radiating body by determining the temperature
of the black body at which the emission intensity in some chosen wave-
length is the same as that of the radiating body. This temperature is the
*' red " or '* green " or ** blue " '* black-body temperature," the exact
wave-length being given in accurate work. According to the method
discussed in the present paper the comparison with the black body is
made on the basis of the same distribution of energy in some limited
region of the spectrum, usually in the visible for convenience, rather
than on the basis of the same emission intensities. Hence the term
** color match temperature." But since in general, precise agreement in
energy distribution of the radiation from a black body and of that from
one of the radiating metals studied can never be secured, the *' color
match temperature " has been defined more accurately as that tem-
perature of a black body at which the relative emission intensities in
some chosen two wave-lengths are the same as those of the radiating
metal under investigation. Numerous experiments showed, however,
that the actual match in color with a black body could be so nearly
obtained for carbon, tungsten, tantalum, platinum, and osmium that the
experimental errors involved in bringing these various substances to a
color match by the use of an ordinary Lummer-Brodhun contrast photom-
eter were less than the errors involved in attempting to bring the various
substances to the same relative emission intensities in two wave-lengths
by the use of a spectrophotometer. Consequently, after establishing
this fact for the various metals to be studied the spectrophotometric
method was abandoned for the more convenient method of " color
match," though for accuracy of conception it must always be borne in
mind that the result accomplished consists in the establishment of a
condition of equal relative emission intensities in some two wave-lengths
near the ends of the visible spectrum — say at 0.5/* and 0.7/*.

As a consequence of the results obtained with this method, and from
other knowledge of the radiating properties of these metals it was con-
cluded that quite probably the ** black body color temperature " of any
of these metals would be higher than the true temperature. If this be
true then the color temperature and the *' black body temperature "
obtained in the customary way, and which we shall hereafter designate^

* The "color match temperature" is also a black body temperature and so it becomes
necessary to designate more precisely the so-called "black body temperature" defined in the
customary way. It is proposed, therefore, to designate the latter as "black body brightness
temperature," or more briefly, "brightness temperature," giving the wave-length where
necessary, and to designate the former as "black body color temperature," or more briefly,
"color temperature," giving the two wave-lengths in cases where an integral color match
cannot be obtained, or is theoretically insufficient.

^^'] COLOR TEMPERATURE SCALES. ^97

the *' black body brightness temperature," or, for the sake of brevity,
** brightness temperature " would give two limits between which the true
temperature would lie, and since the latter is so difficult of measurement,
the ascertainment of the upper and lower limits would give valuable in-
formation. Moreover the simplicity of the process of determining the
color temperature suggested the advisability of using this method to give
a number to the temperature of radiating metals instead of the older and
more commonly used method of determining the brightness temperature
which involves more elaborate apparatus, and, in the case of filaments
of small diameter, is subject to possible large errors. If suitable apparatus
is available, and proper precautions are taken, the brightness temperature
may be obtained with greater accuracy.

In the present experiments the color temperatures of tungsten and car-
bon are determined, and comparison is made between color temperature,
brightness temperature and true temperature of tungsten, using Worth-
ing's^ data for the latter, and between color temperature and brightness
temperature of carbon. The relation is also determined between color
temperature and lamp efficiency in lumens per watt, so that it may be
possible to locate the color temperature and also the true temperature
from measurements of lamp efficiency.

In the earlier papers by two of the authors, to which reference already
has been made, the color temperature of tungsten at low voltage was
measured directly against a black body, and color temperatures at higher
voltages were determined by spectrophotometric comparisons. More-
over comparative data were presented on the brightness temperatures
from observations by Waidner and Burgess, but, as will be pointed out
later, these early values were only approximate as the emphasis at that
time was placed on the application of the *' color match " method in the
study of selective radiation. In 1915, Shackelford,* working in this
laboratory, showed that the color of the radiation from the inside of a
helical tungsten filament was not so white as that of the radiation from
the exterior of the helix, even though the temperature inside was at least
as great as that outside. Subsequently Langmuir* published the same
observation and employed the observed data to given an approximate
scale of color temperatures.

Paterson and Dudding^ in a recent investigation assumed that the color
temperatures are approximately the same as the true temperatures and
obtained results which seemed to show that this assumption was not

> Jour, of Frank. Inst., 18 j» p. 417. 1916; Phys. Rev., Ser. II.

•Jour, of Frank. Inst., 180, p. 619, 191S; Phys. Rbv., Ser. II.. 8, p. 470, 1916.

* Phys. Rbv., Ser. II., 7. P- 302, 1916.

* Proc. of Phys. Soc. (Lond.). 27, p. 230, 191 5.

398

£. F, HYDE. P, E. CADY AND If. E. FORSYTHE.

greatly in error, as they were not attempting to work to an accuracy
gresLtitt than i per cent, in temperature.

11. Apparatus and Method.

An outline of the arrangement of the apparatus used to make the
measurements is shown in Fig. i. The furnace shown diagrammadcally
in Fig. 2 was a vacuum carbon tube furnace somewhat similar to one
already described.' This furnace, with different graphite tubes, was used

D D

I I

Fig. 1.
Arrangement of apparatus.

in most of the experiments, although some check measurements at low
temperatures were made, with platinum-wound porcelain and alundum
tube black-body furnaces of the Lummer-Kurlbaum type. The carbon

furnace shown in the figure was
operated from a transformer sup-
plied with 440 volts which was
stepped down to 40 volts. With
this source of supply there were
required from 50 to 100 amperes
through the primary to heat the
furnace to temperatures ranging
from 1600** K. to 2600'' K. The
current through the primary
could readily be varied in small
steps and so the current through the heater tube, and consequently the
temperature of the heater tube, were easily controlled.

Diaphragms, as shown, were very carefully located so that no light
reached the photometer except from the central diaphragm. This
central diaphragm was made as thin as possible, being only a fraction of
a millimeter thick at the central part. All diaphragms were so cut along
the outer edge that they touched the heater tube only along two V-shaped
edges.

The furnace was so mounted that observations could be made through

> Astrophys. Jour. 34* P« 353. ipii.

Fig. 2.

Diagrammatic sketch of furnace. A. Heater
tube. B. Limiting diaphragm.

)535~^] COLOR TEMPERATURE SCALES, 399

the diaphragm at either end. Inasmuch as the heater tube was mounted
symmetrically inside the container, and the diaphragms inside the
heater tube were equally spaced on each side of the central one, there
was no reason to expect differences in temperature or blackness between
the two ends.

The temperature of the central diaphragm was measured by means of
a laboratory form of the Holbom-Kurlbaum optical pyrometer, directed
toward one end of the furnace while a color match was being determined
for the light coming through the other end. Thus changes in the tem-
perature due to slight changes in the heating current could be detected
and corrected for. In order to be sure that the temperatures at the two
sides of the diaphragm were the same, measurements were made on the
temperature at each side with different optical pyrometers, before and
after each set of measurements on the color. In no case was a larger
difference found than 2° or 3® C. The pyrometers were calibrated, using
a platinum-wound black-body furnace held at the temperature of melting
palladium taken at 1828® K. Extrapolations for the higher temperatures
were made by means of Wien*« equation using sectored disks. Two
thicknesses of red glass (6.8 mm.) (Rotfilter No. F 4512) were used before
the eyepiece of the pyrometer. The effective wave-lengths of the red
glass were obtained in a previous investigation.^ The temperature scale^
used was based on the following values:

Melting point of gold = 1336® K.,

Melting point of palladium = 1828° K.,

C2 = I4»350m X deg.

In making the color temperature determinations the integral light from
the furnace was matched in color with that from a comparison lamp
using a Lummer-Brodhun contrast photometer and the black body color
temperature was transferred to the test lamps by the substitution method.

In making a set of measurements a reading was first taken in the neigh-
borhood of 1800° K., then at higher points and at the end again in the
neighborhood of 1800° K. If the first and last readings were in good
agreement it was assumed that working conditions were satisfactory.
In the early part of this work the only diaphragms in the heater tube
were central diaphragms and the diaphragms near the end. While no
very great differences were obtained when the heater tube was used as
shown, greater weight is given to the later results.

* Aatrophys. Jour., 42, p. 294, 191 5.

« The reasons for the adoption of this temperature scale wUl be published in the October
number of the General Electric Review.

400 E. P, HYDE, F, E. CADY AND W. £. PORSYTBE,

The brightness temperatures of the lamps were determined directly
with the same optical pyrometer tising a very large magnification. This
temperature for the tungsten lamp was also measured by color matching
the lamps under investigation with a tungsten lamp having a large
filament (0.25 mm. diam.), then measuring the brightness temperature
of the large filament. The values obtained by the two methods checked
very well but the latter gave the least variations in the result. The
values found for the brightness temperature of the tungsten filament
as a function of the mean horizontal candles per watt, check well within
the limit of error with results on the same relation obtained previously.*
The values for carbon do not show as good an agreement, due no doubt to
the fact that in the previous case treated carbon filaments were used,
while in this work untreated filaments were used. It was not possible
to get untreated carbon lamps with sufficiently imiform filaments.
Consequently the brightness temperatures of the carbon filaments were
measured in several places and the mean taken.

The comparison lamp for most of the measurements, and three of the
test lamps were 40-watt, no- volt, drawn-wire, vacuum tungsten lamps.
These lamps had their lower supports welded and at the upper supports
the filament was held taut by a coiled spring. The other test lamp was a
lOO-watt, i2o-volt tungsten lamp of the old type having welded lower
supports, and is the same lamp used in previous investigations.' In
some check measurements a 25-watt tungsten lamp of the same type as the
40-watt lamp was used, and a 30-watt, no- volt anchored-oval carbon
filament lamp was used as a comparison lamp. In determining the bright-
ness temperatures a single-loop vacuum tungsten lamp was used, made
with a filament of 10 mil (0.254 mm. diameter) wire and about 30 cm.
long in a bulb approximately 5 inches (12.7 cm.) in diameter.

III. Correction Determinations.

The glass serving as the window of the furnace was not entirely non-
selective in its transmission, thus introducing a possible source of error.
As at all times the substitution method was employed the error was
practically avoided by inserting the same glass between the photometer
and the test lamp when transferring from the comparison lamp to the
test lamp. Hence this source of error was not directly determined and
is not included in the applied corrections (see Table I.).

The selective absorption of the lamp bulbs was corrected by measure-

> Phys. Rbv.. 34, p. 333, 191 2. (Correction must be made to the temperature scale
used in the present investigation.)

* Jour, of Frank. Inst., i6q, p. 439, 1910.

VouX.
Na

X.J

COLOR TEMPERATURE SCALES.

401

Table I.

ReUUion between Lumens per Watt and Color Temperature for a Tungsten Lamp.

Lumens per Watt
(Uncorrected).

Color Tempermtnre
(Uncorvected).

Lumens per Wstt
(Corrected).

Color Temperature
(Corrected).

0.5

1644

0.58

1663

1.0

1777

1.14

1794

1.5

1866

1.70

1883

2.0

1939

2.26

1955

2.5

1998

2.82

2014

3.0

2050

3.37

2066

3.5

2096

3.93

2112

4.0

2138

4.48

2153

4.5

2175

5.02

2190

5.0

2208

5.57

2224

5.5

2241

6.12

2257

6.0

2269

6.66

2285

6.5

2299

7.21

2315

7.0

2327

7.76

2343

7.5

2354

8.30

2370

8.0

2380

8.85

2397

8.5

2406

9.39

2423

9.0

2431

9.94

2449

ments of the transmission for different wave-lengths with a spectro-
photometer. This correction expressed in terms of the change in color
temperature amounted to about 6® at 1800® K. The liunens per watt
of the lamps were corrected for the cooling effect of the leading-in and
supporting wires.^ A correction to the measured lumens was made also
for the absorption of the lamp bulbs. These corrections may be evalu-
ated from the data on uncorrected and corrected color temperature and
lumens per watt as given in Table I. (which see).

IV. Experimental Results.

(a) Color Temperature versus Lumens per WaU, — In order to show the
agreement among the various observations all of the observed points are
plotted in Fig. 3, in which the coordinates are color temperature and
voltage of r — I, a 120- volt, lOO-watt tungsten vacuum lamp which
has been used in previous experiments involving color temperature. It
is seen that for the most part the points lie within 5® of the curve, the
worst deviation being 16®.

Before presenting the experimental results obtained on the relations
between the color temperature, the brightness temperature and the true
temperature of tungsten, it seems best to give the observed relation
between the color temperature of tungsten and the lamp efficiency,

> Trans. lUum. Eng. Soc. (U. S.), 6, p. 338, 191 1.

402

£. P. HYDE, P. B, CADY AND W. B, FORSYTHE.

rSsOOMO

ISviss.

2400*

2300

2200

2100

2000

1900

z8oo

1700

1600

^^^— ^.^p^ ^M>» ■^>^_i w^B^ B^^aa n^aa^ s^M^ iBK^Bi l^^^n BMaiBl i^as^ a^a^ _^^C -^^_a

^i2

M^B^HMi m^^m^ -^^^M^ B^^^i^ H^H^^ Ma^^^H wm^^imm ^^m^^m m^i^mm ^^^m^ ^t^m^^ b^^^bi ^p^^^ imh^pmh a^i^^Ba

*'*^~^" H^^HMM MMMM^HH ^^HMI^ HM^^^ MHM^Ml ^^m^mm ^^^M^ ^^^^Bl ^^H^MH ^^^HMI iV^B^^ ^H^^^H ■^^■WM iMi^^»w

40 50 60 70 80 90 100 no 120 130

Volts
Fig. 3.

The relation between color temperature and volts of a certain tungsten lamp, showing
agreement of observed data.

expressed in lumens per watt. The results are shown in Table I. and Fig.
4. This relation is independent of the size and shape of the filament,
provided it is operated in a vacuum. The lumens per watt of a lamp are
relatively easily determined with a moderately high accuracy, and so
by establishing the relation between the lumens per watt and the color
temperature the latter may readily be found for any tungsten lamp if
the various corrections for bulb absorption and conduction losses are
known. Since these latter, though of the same magnitude for different
lamps of the same general type, are still somewhat different for different
bulbs and for filaments of different size, the results are given in the
original uncorrected values, and also corrected for all these sources of
error. Any investigator may determine the magnitude of these correc-
tion factors for any lamp he may study, or, if so great an accuracy is not
required, he may assume that the corrections are sensibly the same for
his lamps as for those used in the present investigation, and he may
therefore use the uncorrected values.

Vot-X-l
Na4. J

COLOR TEMPERATURE SCALES.

403

In the experiments described, four tungsten lamps were carefully
color-matched with each other at various points throughout the range.
It was found that a definite change in color temperature corresponded to
the same relative change in voltage for each lamp within the errors of

2400

•^

t»'»t\t\

'^^

3300

Q9/\A

«« W

^..

<9T/\/%

SlUil

2000

^ ,

/ >

1900

1800

•

■ >

1700

1600

133456789

Lumens per Watt

Fig. 4.

The relation between color temperature and lamp efficiency of tungsten. B, Author's
data. C. Paterson and Dudding's data.

observation. Measurements of the mean horizontal candle-power were
then made with the lamps operating at voltages to give the same color
as that of the carbon standards in terms of which they were being
measured. In order. to avoid the difficulties due to heterochromatic
photometry it was decided to determine the candle-power at other
voltages by using the table published by Middlekauff and Skogland.^

In Fig. 4, for comparison, the corresponding uncorrected curve for
tungsten obtained by Patterson and Dudding is drawn as a dashed line.
In the neighborhood of i6oo°-i700° K. the two curves are in as close
agreement as could be expected in view of the disclaimer of Paterson

1 BulL Bureau of Standards, //. p. 483, 191 5.

404

E. P. HYDE, P. E. CADY AND W. E. PORSYTHE.

and Dudding of an accuracy better than about 20®, but at the higher
temperatures the differences between the two curves are much too large
to be accounted for on this basis, as at 2400® K. the difference amounts
to approximately 85®. In the neighborhood of 2023® K. (1750® C.) the
difference is 30^-40® and if this correction were assumed in determining
from Paterson and Dudding's data the color temperature of platinum at
its melting point, using the tungsten comparison lamp, the melting point
would have a color temperature of about 2080® K. (1807® C.) or some
50^-60® above the true temperature, which difference would show beyond
question that the color temperature of melting platinum is definitely
higher than the true temperature, a result which must follow from other
published data on platinum. It may be stated in passing that no
attempt will be made to justify the discrepancy between the value found
by Paterson and Dudding using a tungsten comparison lamp, and that

aioo*

2000

1900

1800

I 1700

1600

■^

Lumens per Watt

Fig. 5.

The relation between color temperature and lamp efficiency of carbon. B. Author's data.
C. Paterson and Dudding's data.

derived from a similar comparison against a carbon lamp. They should,
of course, be the same, but it should be stated once more that these
authors do not claim any greater accuracy than that indicated by the
difference in results in these measurements.

The data on the relation between color temperature and lumens per
watt for untreated carbon are given in their observed, or uncorrected
form, in Fig. 5. The corresponding curve obtained by Paterson and
Dudding is also given for comparison, but it should be noted that the
latter curve is based on measurements on both treated and untreated

VOI.X.
Na

COLOR TEMPERATURE SCALES.

405

carbon filaments, whereas the authors' data are entirely confined to
untreated carbon. Owing to the irregularities in the untreated carbon
filaments the accuracy attainable is not nearly so great as that possible
with tungsten, and this fact, together with the consideration of the
relatively smaller errors introduced by end conduction losses, has sug-
gested the inadvisability of attempting to give a corrected curve, as was
done for tungsten.

(b) Color Temperature, Brightness Temperature and True Temperature.
— Reference has already been made to the fact that two of the present
authors^ together with Middlekauff some years ago published results on
the color and brightness temperatures of various metals, giving also the
lumens per watt, but those results will be found somewhat different from
the present values owing to several reasons, principally the lack of

.s
a

u
O

pq 0

m
9

40"
20

20
40
60
80
100

140
180

220

—

—^

■^

•^

1700'

2300

1900 2100

True Temperature. Degrees K
Fig. 6.
The relation between color, true, and brightness temperatures of tungsten.

knowledge at that time of various correction factors to be applied. Thus
the values of color temperature given were for the most part determined
from energy-distribution curves obtained by means of the spectrophotom-
eter, and the slit-width corrections in spectrophotometric measurements
had not then been investigated.* Moreover the brightness temperatures
given in the earlier publication were intended only as approximate values,
as was stated in the paper.

' See references z and 2. p. 395.
Attrophyt. Jour., J5» P* 337, 1912.

4o6

£. p. HYDE, P, E. CADY AND W, E, FORSYTHE.

fSBCOMD
LSUUBS.

The results obtained in the present investigation are shown in Figs. 6
and 7. In Fig. 6 the differences between the true temperatures^ and the
color and brightness temperatures for tungsten are given, the color
temperature values being corrected for the various errors enumerated
in an earlier section. The observations bore out the expectation that the
color temperature would be larger than the true temperature for all
temperatures, whereas the brightness temperatures are of course con-

8
1

■— '

''^

0 10

■^

1700 1900 2100

Brightness Temperature. Degrees K

Fig. 7.
The relation between color and brightness temperatures of carbon.

siderably smaller. The brightness and color temperatures are seen to
give upper and lower limits to the true temperature, with the difference
from the true temperature considerably less in the case of the color
temperature. This general result would very probably be found to
obtain for a number of metals, since available data on the reflecting
powers and also other indications of the optical properties of many metals
show quite general selective absorption of the shorter wave-lengths.

It should be noted that the difference curve between color temperature
and true temperature, if prolonged toward lower temperatures, would
apparently cross the axis, indicating that the color temperature would be
lower than the true temperature if the comparison were made at suf-
ficiently low temperatures. There is no physical reason to believe this
to be true. It is far more probable that the shape of the difference curve
is subject to modification. In the first place it must be emphasized that
experimental errors may occur as large as 5° and possibly somewhat larger.
In the second place differences in the emissivities of different samples of
tungsten may modify this difference curve slightly, since the true tem-
perature and the color temperature are obtained independently from the
brightness temperature, using different lamps.

Direct measurements of the relative intensity of emission in the red

^ Jour, of Frank. Inst., 181, p. 417, 1916; Phys. Rev., Ser. II.

X^^] COLOR TEMPERATURE SCALES, 407

and in the blue, with an optical pyrometer, calibrated in brightness at
the two wave-lengths by comparison with a black body, give a color
temperature curve which differs slightly from the color temperature curve
obtained by integral color match, and indicate a difference curve between
the color temperature and the true temperature more nearly of the form
one might expect. But since the difference between the curves is quite
probably within the experimental accuracy, and since, moreover, there
is always the possibility that with so selective a radiating body as tungsten
the integral color temperature may differ slightly over a wide range of
temperature from the color temperature determined from any two
chosen wave-lengths (approximately 0.665/4 and 0.467/t in these experi-
ments), it has seemed advisable to adhere to the observed curve as given
in Fig. 6, since this is the curve of more practical value.

Independent measurements with a spectrophotometer, and computa-
tions from observed data on the brightness of a black body (to be pub-
lished shortly) both give color temperature scales in substantial agree-
ment with the observed scale (Fig. 6).

The results for carbon are given in Fig. 7. As stated previously the
data on carbon do not justify any attempt to apply corrections similar
to those determined for tungsten. In the case of carbon, since there are
no reliable data on the true temperature, the differences between the
brightness temperatures and the color temperatures only are plotted.

(c) Relation between Color Temperature and Wa4ts, — ^Various attempts
have been made to determine the exponent P in the assumed generalized
form of the Stefan-Boltzmann law for metals,

£ = (tT^.

It is of interest to inquire into the possible existence of a similar relation-
ship between the total radiation and the color temperature. If the color
temperature of a lamp is known at some one wattage and if a simple law
is found to hold for the relationship between color temperature and
wattage, the application of this law affords a convenient way to establish
the entire color temperature scale. The investigation of this relationship
is interesting also because its consideration in conjunction with that of
other established relationships furnishes a check of the original observed
color temperature scale.

Paterson and Dudding give for the above relationship for carbon
lamps (including both untreated and flashed filaments)

WaT^,
and for tungsten lamps

indicating a constant exponent for both metals.

4o8

E. P. HYDE. P. E. CADY AND W. E. PORSYTHE.

Measurements by the present authors, of watts vs. color temperature
(uncorrected) for tungsten and carbon, with the corresponding computed
values of the exponent fi at different regions of the total temperature
interval investigated are given in Tables II. and III.

Considering the data for tungsten (Table II.) it is seen that there is a

Table II.

Relation between Color Temperature and Relative Watts for a Tungsten Lamp Including Corre-
sponding Values of fi.

Color Temperature
(Uncorrected).

ReUtive Watts.i

Computed from Datal

on Color Temperature

and Watts.

Computed from Other
Data.

1750

22.3

4.99

4.93

1800

25.7

4.98

4.90

1850

29.3

4.79

4.87

1900

33.4

4.89

4.84

1950

37.8

4.80

4.81

2000

42.6

4.77

4.79

2050

47.9

4.68

4.76

2100

53.6

469

4.74

2150

60.0

4.82

4.72

2200

66.9

4.73

4.70

2250

74.5

4.79

4.68

2300

82.6

4.72

4.67

2350

91.5

4.71

4.65

2400

100.9

4.66

4.64

2450

110.2

4.26

4.62

Average

4.75

^ 100 « watts corresponding to i.a watts per mean horizontal candle.

Table III.

Relation between Color Temperature and Relative Watts for an Untreated Carbon Lamp Including

Values of fi.

Average

Color Temperature

(Uncorrected).

ReUtive Watts.i

1650

38.9

1700

43.8

4.03

1750

49.2

4.01

1800

55.1

4.06

1850

61.5

3.95

1900

68.4

3.97

1950

76.0

4.11

2000

84.2

4.00

2050

92.9

3.97

2100

102.4

4.09

4.02

1 100 " watts corresponding to 4 watts per mean horizontal candle.

S^^] COLOR TEMPERATURE SCALES, 4O9

distinct indication of a gradual decrease in jS as the temperature increases,
although the successive values of P are not always consistent. This
inconsistency is due to slight irregularities in the observed color tem-
perature scale. The average value of fi over the observed range of
temperature is 4.75, differing from the value of Paterson and Dudding
in the direction to be expected in view of the difference between the two
temperature scales. The following considerations show, however, that
the observed indication of a decreasing P with increasing temperature is
verified in fact.

If / represents candle-power, and if Ti and Tt are two color tempera-
tures at any region of the interval but differing from each other by an
infinitesimal amount, then the following relations hold :

*' I r \ ^*'

^3) rriw) =(f;)

From equations (i) and (3)

/ = fik'.

If the subscript " o " is used to refer to a black body at the same color
temperature, then

/o jSo^o

and hence, since fio = 4,

(4)

Now jfe', which is the percentage change in candle-power for one per cent,
change in watts is accurately known for tungsten lamps, uncorrected for
end effects, and the corresponding quantity ko' for a black body is known
by computation^ to within a very small uncertainty. There is, of course,
the error in locating the precise color temperature of tungsten, — the very
error which gives rise to the present uncertainty as to the constancy of P
for tungsten, but the observed color temperature scale is certainly correct
to within an error of the order of magnitude of 10® and an error of this
magnitude would affect k' by only 0.6 per cent, to 0.7 per cent., whereas
the ratio ko/k' varies by 5 per cent, over the temperature interval 1700®
to 2150'' K.

^ABtrophys. Jour., 36, p. 89, 1912.

4IO E, p. HYDE, P, E. CADY AND W. E. PORSYTHE. [^S2

If now l/lo can be determined, substitution in equation (4) will give
values of fi for tungsten. " / " is the exponent giving the relative change
of candle-power of tungsten for a small change in color temperature, and
" Iq " is the corresponding exponent for a black body. In a previous
paper^ by one of the authors a criterion (Criterion I.) was established for
determining the constancy in emissivity of metals in the visible spectrum,
and it was shown that if the emissivity in the visible is constant for some
metal over a given interval of color temperature, then the relative candle-
power of the metal and the black body over that interval of color tem-
perature will be the same, and so for small steps in that interval I/I9 » i.

Although this criterion was apparentiy fulfilled for carbon and tan-
talum, there was found a slight deviation in the case of tungsten, which
since that time has been verified and more accurately evaluated.* The
deviation in the case of timgsten is approximately 1.5 per cent, and
sensibly the same over the temperature interval investigated. Putting
for 1/Iq its value 0.985 and for ko'/k' the values obtained as indicated, the
values of fi may be computed to a fairly high accuracy. Values obtained
in this way are given in the fourth column of Table II. The agreement
between these values of P and the observed values, given in the third
column, is as good as might be expected, and if, reversing the process, a
color temperature scale should be constructed from the computed jS's it
WQuld agree with the observed scale within less than 5®, which is within
the experimental error. The average values of P by the two methods are
in excellent agreement.

The values of fi for untreated carbon are given in Table III. It is
probable that jS is very nearly, if not quite constant for carbon over the
observed temperature interval, and the uncertainty in the observations,
owing to the lack of uniformity in the untreated carbon filaments vitiates
any effort to analyze the results further. As in the case of tungsten, the
average value of jS for carbon is less than the value found by Paterson and
Dudding, but here again this is to be expected in view of the diflference in
the corresponding color temperature scales.

Summary.

The ** black body color temperatures '* or simply the " color temper-
atures " for tungsten and untreated carbon lamps are given from direct
observations against a carbon-tube black-body furnace, plotted against
lumens per watt of the lamp.

For tungsten the differences between the ** color temperature," the

* Loc. dt.

« Worthing, loc. dt.

5^^] COLOR TEMPERATURE SCALES. 4 1 I

'* brightness temperature " (ordinarily called heretofore the " black-body
temperature ")» ^md the true temperature are given, and it is pointed
out that the color temperature is greater than the true temperature,
whereas the brightness temperature, as is well known, is less than the
true temperature. The color temperature, however, is much nearer the
true temperature.

For carbon the difference between the color temperature and the
brightness temperature is given, the true temperature being unknown.

The relation between the color temperature and the watts is investi-
gated, and found, within observational errors to obey approximately an
exponential function. It is shown that for tungsten the exponent cannot
be a constant, but must decrease slightly from to low high temperatures.

412 HEW BOOKS. [:

NEW BOOKS.

Radiodynatnics. By B. F. Miessner. New York: D. Van Nostrand Com-
pany, 1916. Pp. V + 206. Price, {2.00.

With the entry of the United States into the war general interest in the
subject matter treated in "Radiodynamics" by B. F. Miessner has greatly
increased. In the opinion of the reviewer the book should be in the library
of any physicist who is at present interesting himself in the problems which
our entry into the war has brought to the fore. ' The general subject treated
is communication by means of radiations and control of moWng mechanisms
by the same means. There is no attempt to explain principles which can of
course be studied quite easily from other sources and which are well known
to the readers of the Review.

The value of the book lies in its being an up-to-date statement of what has

been accomplished and the means employed in the control of mechanisms by

means of the various forms of radiant energy.

A. T.

Telephone Apparatus. By George D. Shepardson. New York: D. Apple-
ton and Co., 191 7. Pp. xvii + 337-

This book was written to supply the need for a systematic historical and
theoretical treatment of the subject. It will be found useful not only to the
telephone engineer, but to the physicist who wishes to keep in touch with the
applications of his science. Part I. reviews the fundamental acoustical prin-
ciples and describes the various types of transmitter and receiver. Part II.
discusses signalling equipment. Part III. treats of sources of electromotive
force, the uses of condensers and induction coils, and protective devices.
The scope of the book does not include the consideration of telephone circuits
or of wireless telephony. The descriptions of some of the many devices referred
to are necessarily somewhat meager, but there are abundant references to other
books and to periodical articles. Several appendixes are devoted to the
laws of the magnetic circuit and to a more extended development of the
mathematical relations employed in the text. The book is well printed and
illustrated, and is provided with good author and subject indexes.

E. P. L.

Atoms, By Jean Perrin and translated by D. L. Hammick. New York:
D. Van Nostrand Co., 1916. Pp. xiv + 211. Price, J2.50.

This translation of the fourth revised edition of this little book will be
welcomed by all English-speaking physicists and chemists. The topics treated
are: Chemistry and the atomic theory, including the phenomena of solution;

f&:^'] NEW BOOKS, 413

molecular agitation and kinetic theory; the Brownian movement, including a
detailed account of the author's work and of Einstein's theory; phenomena due
to fluctuations of density in liquids and gases; the radiation of black bodies,
Planck's theory of quanta, and Einstein's theory of specific heat: the atom of
electricity and the methods of measuring the elementary charge; the genesis
and destruction of atoms as manifested in radioactive phenomena. It is to
be regretted that no account is given of spectroscopic phenomena which are
closely related to atomic theories, such as the Zeeman and Stark effects, and
X-ray spectra, or of the work on atomic numbers, or of the theories of atomic
structure which have recently attracted so much attention. In spite of these
omissions, no other single book known to the reviewer gives such a compre-
hensive survey of this important field. Although necessarily concise, the
treatment is by no means superficial.

An erroneous reference on page xiv to the Rice Institute as the University
of Houston has not been corrected by the translator. On page 22 the numbers
653 and 65.3 should be 65.7. On page 49 " J cent'tniUiardihne^* is translated

"the hundred-thousandth."

E. P. L.

Radioactivity, By Francis P. Vbnablb. New York: D. C. Heath and Co.,
1917. Pp. vi-f54. Price $.50.

This little book is the outcome of work which originally was given in lecture

form to students of elementary chemistry. It should prove useful to students

of science who have not the time or opportunity for reading the more complete

treatises on the subject. While the book is very small, and can be read at a

single sitting, yet it contains in a well-arranged and clearly written form the

most important results of the work done in this field. The text is unusually

free, for so popolar a treatment, from errors and misleading statements.

Teachers of physics and chemistry will find it a good book for reference for

their beginning students.

O. M. S.

X'Rays. By G. W. C. Kaye. Second Edition. London: Longmans Green
and Co., 191 7. Pp. xxi+285. Price, $3.00 net.

The author does not claim to have written "a treatise or hand-book on

X-rays." The book contains brief accounts of some (not all) of the important

papers that have been published on X-rays up to the middle of 1916. The

second edition does not differ very much from the first. Unfortunately a

great deal of material that is now known to be inaccurate, and even incorrect,

is described in considerable detail, whereas some of the most important re*

searches are eicher not mentioned at all, or referred to in a few words. In

spite of this the book may be useful to the student of X-rays in that it provides

a concise statement (with references) of some of the research work that has

been done on X-radiation.

W. D.

414 NEW BOOKS. [i

The Theory of Measurements, By Lucius Tuttle. Philadelphia: Jefferson
Laboratory of Physics, 1916. Pp. xiv+303. Price, $1.25.

Written in the form of a laboratory manual with extended discussions,
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tities. The author discusses general considerations of experimental procedure
and measurements, the theory of error, accuracy, adjustment of observations,
graphical methods, use of slide rule, logarithms, etc. The mathematical
knowledge required is no greater than that usually required of a student in a
college course of general physics. Each step is introduced by an appropriate
laboratory exercise which offers a definite objective illustrat'on of the matter
under discussion. This plan of procedure will in no way hinder the use of the
book by a reader who can not perform the suggested experiments. In fact
it probably adds to the clearness of presentation and improves the treatment
for the average reader.

Nowadays it is the tendency to point our laboratory work towards a great
emphasis of the principles of physics with the probable result that we overlook
often the theory of the means which we employ, the theory of quantitative
observation and computation. Hence a book like this can well be used to
supplement regular courses in laboratory physics. Part of the material found
here is in some laboratory manuals, but in general with a treatment very
incomplete. It is a serious question whether we have not gone entirely too
far in reducing the amount of this kind of work in our elementary laboratory
courses. Certainly this book contains much that every advanced student of
quantitative experimental science should be thoroughly familiar with.

O. M. S.

Physical Laboratory Experiments ^or Engineering Students. By Samuel
Sheldon and Erich Hausmann. New York: D. Van Nostrand Co., 1917.
Pp. V+I34-

This book was prepared for the use of sophomore students in the Polytechnic
Institute of Brooklyn. Part I. contains thirty experiments on mechanics,
sound, heat and light, forty illustrations and diagrams, and an appendix in
which are given tables of physical constants. The book is especially adapted
for a laboratory course for students who have had some previous knowledge of
physics and mathematics including the calculus, but who do not intend to take
an advanced laboratory course In physics. The illustrations are of modern

apparatus manufactured by well-known makers.

E.J.

Second Series. November, IQ17. Vol. X., No. 5

THE

PHYSICAL REVIEW.

OSCILLATING SYSTEMS DAMPED BY RESISTANCE PRO-
PORTIONAL TO THE SQUARE OF THE VELOCITY.

By J. Parkbr Van Zandt.

I. Introduction.

LL bodies moving through a resisting medium encounter forces which
affect their motion. In order to predetermine the motion of any
dynamical system it is necessary to know the factors influencing the re-
sisting forces and the laws governing their variations. The resistance
may influence the motion very markedly; thus in the motion of large
bodies at relatively high speeds, as in the propulsion of aeroplanes or of
ships, the amount and variation of the resistance are dominant factors
and determine the maximum velocity obtainable. A comprehensive
study of the laws of damping is difficult because of the many variable
factors influencing the forces of resistance. Since in many of the more
important problems of engineering the predominant factor is velocity,
we shall restrict the discussion to the variation of the forces of resistance
with the velocity of the moving body. The resistance is considered as
varying only with integral powers of the velocity, although experimentally
it has been shown that the exponent of the resistance term is often frac-
tional. Such a restriction is necessary because it is extremely difficult
if not impossible to treat mathematically the irrational expressions which
would otherwise occur. From a practical standpoint, the introduction
of irrational or fractional exponents is in most cases an unnecessary
refinement because of the presence of other indeterminate factors, parti-
ally independent of the speed, which modify the damping.

It is sometimes assumed that the majority of forms of resisted motion
fall into the class in which the opposing forces are directly proportional
to the velocity. Many important engineering problems, however, involve
resisting forces which are proportional to the square of the velocity.
Thus in hydrodynamics the resistance to the motion of water in pipes,

41 6 J. PARKER VAN ZANDT. [to»

conduits and surge towers, and the resistance to the motion of vessels,
are approximately proportional to the square of the speed. When the
resisting medium is air we have such problems as those of ballistics, of
aerodynamics, of resistance to the motion of trains, and of air damping in
some forms of electrical meters. One of the first to recognize the presence
of the '* square law " of resistance in the motion of projectiles through air
was Sir Isaac Newton.^ He observed the time of fall of spheres dropped
from St. Paul's Cathedral and verified the law for moderate speeds. It
has been found that for speeds approaching the velocity of sound, the
effect of the elasticity of the fluid is felt and the index may rise consider-
ably above its value when the resistance is a pure quadratic function.
Comparing the results of German, Dutch, Russian and English experi-
menters, Cranz* has found that the index increases from 2 to 5 and then
decreases to 1.55, as the speed of the projectile is increased up to and
beyond the velocity of sound in air.

In aerodynamics the investigation of the laws of air resistance for large
bodies moving at relatively high speeds has been for several years one of
the chief occupations of aeronautical research. Zahm,' Lanchester* and
others have studied the problem and find two factors, one called the head
resistance, which varies with the square of the velocity within certain
limits, and a second form of resistance, caused by surface friction, which
varies with the 1.86 power of the velocity. It is of interest to electrical
engineers to know that formerly some of the electrical supply meters
depended for their operation upon the motion of fans in air or liquids,
which caused resisting forces varying with the square of the speed. Such
were the Forbes, the Schallenberger, the Ferranti and the Slattery
meters.* A close similarity exists between the action of the retarding
forces in air and in water. It has been shown experimentally that the
exponent for surface friction when water flows through pipes and channels
is 1.85; this is almost exactly the value given by Zahm for the resistance
of the air when flat or tapering bodies move edgewise. If the flow of
the water is turbulent the index rises to the square,* corresponding thus
to the expression for the resistance met by blunt bodies propelled through
air. In viscous fluids, Stokes's law of resistance varying with the first

» Sir Isaac Newton, Principia, Book II.. Sec. VII.

* Cranz, "Ballistik/* Encykl. der math. Wissenschaften, Vol. IV.. Part 2, 1903. Contains
a complete bibliography. See also Encycl. Brit.. Ballistics.

*Zahm. Philosoph. Soc. Washington. 1904.

^Lanchester, Aerial Flight. Vol. I., Chap. 2. 1908; A discussion concerning the Theory of
Sustentation and Expenditure of Power in Flight. Eng. Cong.. San Francisco, 191 5; Berriman,
The "Arrival" of the Aeroplane. Eng. Cong.. San Francisco. 1915.

* Swinburne. The Measurement of Electrical Currents. 1893; Parr. Electrical Engineering
Measuring Instruments. 1903.

* Knibbs. Proc. Roy. Soc. N.S.W.. Vol. 31. 1897.

JJSJy^] oscilla ting systems, 417

power of the velocity, based on the assumption of non-sinuous motion
with no slipping at the boundary, was found by Allen* to hold for the
motion of bubbles and small solid spheres in liquids if the velocity re-
mained smaU, but if the velocity was increased sufficiently the resistance
approximated the square law closely.

An interesting problem in hydraulic engineering appears in the design
of surge towers to regulate the oscillations occurring in long conduits
when the rate of flow is suddenly varied. If the flow is shut off entirely,
the surges arising between the tower and the adjoining conduit represent
in effect a freely oscillating system damped by resistance approximately
proportional to the square of the velocity. In most of the papers listed
below* a graphical or step-by-step method of integration is presented;
in some the resistance is assumed to vary with the first power of the ve-
locity; in none is a strictly analytical solution given for the oscillations
when the resistance is assumed proportional to the square of the velocity.
The resistance to the motion of vessels has been found to be similar to
that discussed in the paragraph on aerodynamics. The head resistance
varies with the square of the speed while the frictional resistance is pro-
portional to a somewhat lower index. For the case of a ship rolling in
still water White^ assumed the resistance to vary partly as the angular
velocity and partly as the square of the angular velocity. When rolling
among waves the motion of the ship becomes a forced oscillation with a
periodic forcing cause and damped by resistance varying with both the
first and second powers of the angular velocity.^

This brief discussion is in no sense complete; it is intended merely to
suggest the wide scope and prevalence of the square law of damping, and
to indicate certain problems involving oscillating systems which are
presented for solution.

11. Free Oscillations Damped by Resistance Proportional to

THE Square of the Velocity.

The equation of motion for a system executing free oscillations and
opposed only by a force varying as the square of the velocity is

^ Allen, Phil. Mag., Scr. 5, Vol. 50, 1900, pp. 323 and 519.

« Johnson, The Surge Tank in Water Power Plants, Trans. Am. Soc. Mech. Eng., Vol. 30,
p. 443, 1908; Uhl. Speed Regulation in Hydro-Electric Plants, ibid.. Vol. 34, p. 379; 1912;
Durand. On the Control of Surges in Water Conduits, ibid.. Vol. 34, p. 319, 191 2; Warren.
Penstock and Surge Tank Problems. Proc. Am. Soc. Civ. Eng., Vol. 40. 2, p. 2521, 1914; Church,
Surge in a Hydraulic Stand Pipe. Cornell Civ. Eng., Dec., 191 1; Forcheimer, Zeitschr. d. Ver.
Deutsch. Ing., Vol. 56, p. 1291, 1912 (includes a bibliography of the foreign literature); Prasil,.
Surge Tank Problems, The Canadian Eng., 1914 (includes a bibliography).

» Sir Wm. H. White, Trans. Inst. Nav. Arch., 1895.

«Sir Philip Watts, Shipbuilding, Ency. Brit., 1910. See also Henderson. Engineering,.
April 18, 1913-

4l8 /. PARKER VAN ZANDT. [i

5^±iv(^)+jf« = o. (I)

Poisson^ and others have investigated this equation for small oscillations,
considering the higher powers of the successive arcs as negligible. Routh*
performs the first integration and obtains a relation between the velocity
and the displacement. GrammeP and Ignatowsky^ give approximate
solutions showing the displacement in terms of the time, useful when the
oscillations are small but not adapted to larger values of the displacement.
In the following pages a solution is developed which is particularly
adapted to large displacements and heavy damping. In addition an
experimental method is presented, by which it has been found possible
to verify the results of the analytical study.

Transforming equation (i) by means of the proper substitutions and
performing the first integration, the expression for the velocity in terms
of the displacement is

=F-d + Ci€'*^, (2)

2iV* N

wherein Ci is the constant of integration and depends on initial conditions.
In order to define the meaning of the remaining symbols let us consider
the motion of a torsion pendulum to be thus expressed. Then 6 is the
angle of displacement from the position of equilibrium; M is the restoring
torque in cm.-dynes per radian of twist, divided by the moment of
inertia of the pendulum about the axis of support; N is the coefficient of
resistance, that is, the torque per square of unit angular velocity divided
by the moment of inertia. The double sign before N is necessary because
a change in the direction of the motion does not automatically change the
sign of the resistance term. At each turning point, therefore, a discon-
tinuity is introduced into the equation.

The integral of equation (2) expresses the relation between the angle
of displacement 6 and the time T.

It is not possible to integrate the right-hand member of equation (3) in
the form as written. For certain restricted values of N and 6, however,
it is possible to express this integral in terms of a converging series and

^ S. D. Poisson, A Treatise of Mechanics, Vol. I., Sec. 186-190.

* E. J. Routh, Dynamics of a Particle, Art. 129, 1898.

* R. Grammel, Physik. Zeitschr., Vol. 14, p. 20, 1913.

* W. V. Ignatowsky. Archly, d. Math. u. Phjrs., Vol. 17, p. 338, 1910.

^2J"$f*] OSCILLATING SYSTEMS, 4I9

thus to obtain a relation between 6 and the time. Substituting
2 = (i =F 2N6) and assuming as initial conditions 6 ^ 60 and dd/dt = o
when r * o, in order to determine Cu equation (3) becomes

T- C ^ -

>/2M

/w-"[>-(s)Tr'*- <4>

C is the constant resulting from the second integration and «o is the value
of « at r — o. The upper sign in the expressed value of z corresponds
with positive angular velocity, the lower sign with negative angular
velocity. If the absolute value of (zoe'/t'^z) is less than unity, the right-
hand member of equation (4) may be expanded according to the binomial
theorem into an infinite series which converges absolutely. Assuming
for the moment that the values of N and 6 are such that this expansion is
permissible.

••• + -^ i-ky^ + '"\dy,

where k = 2q/€*» and z =^ y^. The integration of the series, equation
(5), may be expressed as the sum of the integrals taken term by term.
The general integral is

(6)

For values of Zot'/t'Ki less than 0.99 it is sufficient for most purposes to
consider the first six terms of the binomial expansion, equation (5), and
the first seven terms of the expansion of each integral. The result is the
following equation expressing the time T in terms of the displacement 6,

(7)

where y = ^z = v^i =F 2NO. The constants (a, j8, 7, 5, f, • • •) are found
by collecting the coefficients of the terms in like powers of y obtained from
the expansion of each member of equation (5) . A simple relation connects
each constant with the preceding constant. All of the terms in the right-
hand member of (7) are known when the constants N and M are specified
and the desired value of d is chosen. We have therefore a complete
solution of the differential equation (i), subject only to the limitation

420 /. PARKER VAN ZANDT. [i

in the values of N and 6 necessary to make the expansion absolutely
convergent.

If Zq€'/€'»z is greater than unity equation (4) may be written

This expresses the product of two absolutely convergent series. It
may therefore be developed by the method outlined above and a second
solution for T found, in terms of the displacement 6. This solution will
be applicable for all values of N and 0 for which (7) is not valid, save for
the unique case ZQe',U'»z = + i which renders both forms of solution
indeterminate. It will be noted that at 7" = o, 2 = Zo and the solution
assumes the indeterminate form. Hence it is necessary to know the
value of T for any one angle 6 other than 6q, say the value of T when
d = o and the vibrating system is passing through the position of equi-
librium. At this value the series converges very rapidly. The constant
of integration may be derived as well from the value of T at the first
turning point, but here the expression (7) is in general less rapidly con-
vergent.

We have in equations (7) and (8) relations by which the value of the
displacement may be found at any instant. It is often desirable and
sufficient for the purpose at hand to find merely the value of 6 at the turn-
ing points. This may be done by placing dd/dt = o in equation (2) and
determining Ci from the condition that 6 =^ 60 when 7^ = 0. Then if
^1 is the displacement at the first turning point,

(I =F 2Nei)t^^'' = (I T 2Ne^)t^^' s A, (9)

Expanding c*^*» into an infinite series which converges absolutely for
all values of ± 2Ndu equation (9) may be reduced to the following form:

2Nei = ^ Y"7-^ » (10)

where

2j^ (2Ni^ (2Nie,iy (2NeyY ^ . ^

3 8 ' 30 ' 144 '

All of the terms in X are additive, even when the angle di is negative;
it is a rapidly converging series, especially when 2N/61J is less than unity.
In general 2N/61I is considerably less than unity so that the equation
presents a practical form for finding 61 by successive approximations.
It appears also from equation (10) that the ratio of successive arcs is
independent of the magnitude of the restoring torque JIf , contrary to the
relations obtaining when the damping is proportional to the first power
of the velocity.

iSt^] OSCILLATING SYSTEMS. 421

In order to study the period of the oscillations return to equation (2).
If V9 is the velocity of the moving system when 6^0 then Vf?
= M/2N* + Ci and equation (2) may be written:

•"'" (.2)

If the product NO is considerably less than one the expression within the
brackets will not differ much from unity. We may therefore write

— /■

7o« T 2Vo*Ne + 2CiiW]->/*«W

from which

(13)

%^7o^ - 2CtVo* . r_ (M2, ^,1 7o« . ,

If Tq is the period of the oscillation then

^■\-2Nl$J

(51)

(I + 2NN) ' '^

When iV, the coefficient of damping, is zero To = 2ir/v^Jlf, which is the
usual expression for the period of undamped harmonic vibrations. If
2N/6q/ = o.io the period is increased by 0.25 per cent, above that for
undamped motion. This indicates that for moderate values of damping
and displacement the period is lengthened slightly and the motion remains
practically isochronous. If 2N6q approaches a value of unity or greater,
equation (15) is meaningless because the approximation employed is no
longer valid. It may be shown experimentally, however, that for the
larger values of 2N6q the period is a little longer and the motion is still
almost isochronous, just as in the case where the resistance varies as the
first power of the velocity.

If values of 6 are known at the successive turning points of the motion,
by observation for example, an approximate value of iV, the coefficient
of resistance, may be found by a consideration of the energy relations
during any one oscillation. At the initial position the velocity is zero
and all of the energy is potential; this is true also at the next point of
rest ^1. The difference between the amounts of potential energy is
equal to the dissipation of energy caused by the resistance, during the
excursion from So to 6i. Referring again to the torsion pendulum, if r
IS the static moment of force per radian of twist in the suspending wire;

422

J, PARKER VAN ZANDT.

rSMOMP

le is the moment of inertia, and N is the coefficient of resistance as before,

IW - e.^] - - m. £ (f;)* de.

(I6)

If d6/dt can be expressed as an integrable function of 6, this equation will
serve for the determination of N. Let the velocity be plotted in terms of
displacement (Fig. i) from equation (2), assuming for the moment any
reasonable value of N. For moderate values of damping this curve
during any one oscillation suggests a portion of an ellipse, with its center
displaced from the origin of coordinates to the point corresponding to
maximum velocity. That is, the motion is approximately simple har-
monic about a center which is moved first to one side, then to the other
side of the origin. Let 6* be the value of 6 when the velocity is a maxi-
mum, and write a ^ 6 ^ d', then

(t)"-(f)*-z'<«— ''-<-•''•'

(17)

expresses the simple harmonic relation between velocity and displace-
ment. Substituting in equation (16), integrating and solving for N^
its approximate value is

iV=3

1^0 + m

(18)

Fig. 1.

This approximate value may
then be substituted in equa-
tion (10) and by successive
trials a new value of N may
be found which will corre-
spond with the observed turn-
ing points, with any required
degree of precision.

As a check upon the
method of solution elaborated
in the preceding pages, it
may be noted that equation
(2) can be integrated by
graphical means. For exam-
ple, let iV = i.o; Af = lo.o;
$q = i.o radian. If ^ be taken
as the independent variable
and the reciprocal of the ve-

locity be plotted against By the curve shown in Fig. i is obtained. Discon-

Nas. J

OSCILLATING SYSTEMS,

423

tinuity of resistance requires the use of the double sign in equation (2),
and the constant of integration must be redetermined for each successive
excursion. The area of any narrow strip extending from this curve to
the axis of 0 represents the amount by which T is increased, while the
moving system passes through the corresponding angle. Using a plani-
meter, T may then be obtained in terms of 0 by means of a step by step
area summation. Plotting 0 in terms of T, the curve shown in Fig. 2 is
a graph of the motion demanded by the original differential equation.

Plot of -t: =F I

Fig. 2.

.0 ( ~ J + lo.o tf - o

Full Line by Fig. i. Circles by equation (7).

To compare with the results of the graphical integration, the same values
of N, M and ^0 were substituted in equation (7). T was thus obtained
for several values of B. The results are shown as small circles placed
upon the curve, Fig. 2. The agreement between graphical integration
and the computations from equation (7) is apparent.

III. Experiments.

Few attempts have been made to construct models for the study of
oscillations damped by resistance varying with the square of the velocity.
Durand^ has suggested the use of small models of surge chambers and the
extension of the results of observations, by the law of kinematic similitude,
to surge towers of full size. No other reference has been found to any
experimental means of reproducing and studying oscillations of this kind.
It is obvious that any type of pendulum may be used, provided that the
resistance is convertible into a function of the square of the velocity. It

> W. F. Durand, Trans. Amer. Soc. Mech. Eng., Vol. 34. p. 359, 1912.

,QiS£,

424 /. PARKER VAN ZANDT. l^SSS?

will be advantageous to choose a model such that the coefficients of the
resisting and the restoring forces can be varied at will, in which a minimum
of inherent errors and unknown factors is introduced, and for which the
required velocity is not so great as to render accurate observation dif-
ficult. There are doubtless many ways of obtaining an automatic ad-
justment of the resistance. Certain inherent difficulties appear to offset
the advantages of automatic control; in the following pages a method
depending essentially on electromagnetic induction is described, in
which the necessary simplification is obtained by means of a manually
operated device.

An annealed copper disk, 15 cm. in radius, is suspended from a rigid
support by a phosphor-bronze torsion wire. The disk is accurately
leveled and centered and oscillates between the jaws of a small soft iron

electromagnet in such a manner that
xnt»3 rating tim^nrnt eddy curreuts are generated in the disk.

The electromagnet is hung by a silk
thread and hence is free to respond to
the drag of the rotating disk caused by
the Foucault currents. Attached to the
n - Af^irtr 'T— iS I ^•fe. lower jaw of the magnet is a small

5 jl >w r.tiTbhTClw^* t *^ spiral spring which resists the tendency

^"^ to turn, so that the actual displace-

Fig. 3. ment of the electromagnet from its po-

sition of equilibrium dep)ends directly on
the speed of rotation of the disk. The image of a brightly lighted, ver-
tical slit is reflected from a mirror attached to the back of the magnet to
a volt-potentiometer arranged in the form of an arc. The distance of the
spot of light from the center of the arc thus gives a continuous measure
of the velocity of the rotating disk.

The volt-potentiometer consists of a meter stick bent into an arc of
radius 55 cm. and effective length 60 cm. Every half centimeter there
is a commutator segment consisting of a copper wire held firmly against
the face of the stick. Each pair of segments equidistant from the center
is connected through a fuse and switchboard to a supply of four volts
from a large lead cell storage battery. Completing the. circuit between
the potentiometer center and the sliding contact are two pairs of specially
designed damping coils, in the field of which the copper disk oscillates.
Each of the coils has a radius of 6 cm., a wooden core of 2 cm. radius and
is wound with 52 layers of number 20 enameled copper wire. The air
gap in which the disk rotates is just large enough to permit perfect clear-
ance. It is thus possible to obtain very heavy damping. The sliding

Plate I.
To face page 415

Fig. 6.
J. PARKER VAN ZANDT.

VOL.X.

Nas

!"]

OSCILLATING SYSTEMS.

425

Vtotr ^mrmMTtOMm rtm ditto

contact on the volt-potentiometer is kept by hand under the spot of light
reflected from the mirror on the integrating magnet. Hence at any
instant the voltage across the damping coils is directly proportional to
the velocity of the moving disk. Since there is no iron present and the
speed of the disk is moderate, the current through the damping coils
will vary sensibly as the voltage applied. Therefore the field set up by
the coils varies directly as the velocity of the disk. Now the eddy cur-
rents generated in the rotating disk are directly proportional to the
strength of the field, and the torque developed, which opposes the
motion, is directly proportional to the product of the generated eddy
currents and the existing field. Therefore the oscillations of the torsion
pendulum are damped by a resistance varying as the square of the ve-
locity. To regulate the value of the coefficient of damping it is only
necessary to change the maximum voltage applied across the coils; or
the deflection of the spot of light may be varied as desired, by changing
the current actuating the electromagnet.

For the purposes of observation a
circular millimeter scale 40 cm. in length
is placed on the disk and turns with it.
A fixed pointer leading out to the scale
enables the successive turning points to
be read with an accuracy of one part in
four hundred, provided that the period
of the swing is moderately large. If a
continuous record of the displacement

is desired a photographic record of the rotating scale may be taken.
When the torsion wire is initially twisted, a device beneath the disk
clamps the axle in such a manner that the disk may be released without
jar. The torsion wire is joined to the disk by a small cap which is screwed
on the axle, so that a new suspension may be substituted easily. The
model may readily be adapted to show forced oscillations, motion
damped by a combination of first power and second power resistance,
rectilinear damping, and so on. (See Figs. 5-6.)

Considering the inherent errors of the apparatus: (i) There is always
present a small amount of damping proportional to the first power of the
speed. This is due to the drag of the integrating magnet and to the
slight air friction on the upper and lower surfaces of the thin copper disk.
(2) The current flowing in the coils and hence the field set up, is not
exactly proportional to the voltage applied because the electromotive
force is not constant but varying. In all of the measurements the
resistance of this circuit in ohms was at least seventy times the inductance

Fig. 4.

426

J. PARKER VAN ZANDT.

of the damping coils, measured in henrys, and the period of oscillation of
the disk was always greater than ten seconds. It may easily be shown
by integration of the equation of electromotive forces that no appreciable
error is introduced when the current is assumed proportional to the applied
voltage. (3) An effect necessary to overcome was the influence of the
varying field of the damping coils on the suspended magnet. To obviate
this, a small coil was placed between the damping coik and the nu^^net
and connected in series with the coils in such a way that its field almost
entirely neutralized the troublesome stray induction. It was also found
advantageous to maintain a weak general field about the magnet in
order to reduce the importance of stray magnetic fields.

The copper disk weighing 755.5 grams was suspended by a No. 22
phosphor-bronze torsion wire, 35 cm. in length. From the change in the
period due to the addition of a turned brass cylindrical disk weighing
976.5 gm. and also by direct computation from dimensions, the moment
of inertia about the axis of support and the constant of the torsion wire
were found; by means of these values the coefficient of the restoring
torque, Jf, was found to equal 0.294 c.g.s. imits. The natural, or
undamped period, measured by chronograph and stop-watch, was 11.604
seconds. A current of 0.60 amperes was then, sent through the electro-

Fig. 7.

magnet circuit, a maximum voltage of 216 volts was applied across the
potentiometer, the disk was displaced initially through 360 degrees, and
turning points were observed. The motion is plotted in Fig. 7. The
unknown coefficient of damping was found approximately, N = 0.0370.
This value of N was used to determine from equation (10) the values of
successive turning points, under the assumption of a resistance varying
solely with the square of the velocity. The values are shown in Fig. 7 by
circles. The effect of the small unavoidable damping, varying with the
first power of the velocity, is apparent, the first, second, third and fourth

5^ jf •] OSCILLA TING S Y STEMS. 427

observed turning points lying a little inside of the computed values. It
was found that the greatest variation from pure square damping between
any two consecutive oscillations was 2.5 per cent. This is within the
anticipated minimum of unavoidable error.

The next step was a more detailed comparison of the analytical solu-
tion, equation (7), with an experimental system. The restoring torque
was maintained at its former value, but the coefficient of damping was
increased by means of larger current in the damping coils. From several
observations the first three turning points were found to be 360®, 197**,
149®. The new coefficient of damping was computed approximately
from equation (18) and more precisely from equation (10). N = 0.0960.
The constant of integration for equation (7) may be computed if we know
the period of the oscillation. Experimentally the half-period, or the
time for the first swing, was found to be 5.860 seconds. It may also be
determined analytically from the relation

T^M = TyUiiK), (19)

where T, M and Tu Mi are the half-periods and coefficients of the re-
storing torques, respectively, for two independent systems and (K) is
the ratio of their correction factors (see equation (15)). Let us take Tu
Ml for the case solved by graphical integration, Fig. 2; then T = 5.840
seconds, which is in close agreement with the experimental determination.
All of the terms in equation (7) are now known. It may therefore be
applied to the first oscillation in the manner already outlined. The
graph of the motion is plotted in Fig. 8 and the values as computed by
equation (7) are indicated by circles. It will be seen that the agreement
IS entirely satisfactory. For the second half-period, equation (7) is not
applicable because the original expansion is not convergent. To deter-
mine the motion for the second oscillation analytically, it would be
necessary to employ the expanded form resulting from the development
of equation (8).

Figs. 7 and 8 exhibit the motion of the oscillating systems as practically
isochronous. We have proved analytically that this is true for moderate
damping. (See equation (15).) It may also be shown experimentally.
A smoked drum chronograph 50.5 cm. in circumference was driven by a
small motor connected through a friction drive and reduction gears.
Two needle pointers mounted on a traveling screw traced continuous
paths on the drum. One pointer was actuated by a contact on the pen-
dulum of a laboratory clock so that it indicated seconds. The other
pointer, by means of a tapping key, was made to record the successive
instants at which the torsion pendulum passed through its position of

428

J. PARKER VAN ZANDT,

[SSCOND

equilibrium. For the motion of the oscillating system plotted in Fig. 7
five independent records of the times for successive swings were taken
and the first nine average values were found to be 5.81, 5.85, 5.77, 5.87,
577» 5-83» 578, 5-8i, 582 seconds. The average half-period was a little
more than 5.81 seconds and the greatest deviation from the average was

aio

Of ,

ao9(

f a2 H0

' 0

Cor.

iput

0
•

•

Til

i« 1

5«c<

tn€l4

il2

org

■ II

Fig. 8.

0.06 second, or approximately one p)er cent. This is well- within the
expected accuracy, since it is probable that for any single observation
an error of one tenth of a second would easily enter in tapping the key a
little too soon or too late. We are therefore furnished with experimental
evidence that for moderate and even heavy damping the oscillations are
practically isochronous.

IV. Use of the Experimental Model in the Study of Other Forms

OF Damped Oscillations.

It has been mentioned that the model described in Part III. is capable
of extension to other problems in damped harmonic motion. Of these
problems, three have been given some consideration, experimentally and
analytically. These are: (i) free oscillations subject to both first and
second power damping, the first power predominating; (2) forced, con-
tinuous vibrations; (3) rectilinear damping. A detailed discussion of
each case is beyond the scope of this paper. We shall merely point out
the lines along which the investigation has proceeded.

Vol. X.
No

^^'] OSCILLATING SYSTEMS. 429

1. In order to produce a resisting torque due to both first and second
power damping it is only necessary to add to the existing arrangement a
constant damping field. By means of a small coil with an iron core (de-
magnetized before each setting), through the field of which the disk
rotates, a wide range of damping is readily obtained. (See Fig. 6.) One
feature of the greatest advantage lies in the ability to study each of the
two components of the damping singly as well as combined. We may
therefore make an independent determination of each coefficient of damp-
ing apart from the more complicated motion due to their joint action.
Analytically the oscillations may be treated by the double approximation
method of Routh,^ provided that the first power damping predominates.
The following conclusions have been established experimentally for
systems having an initial displacement of 27r radians: (a) the oscillations
are isochronous; (6) the period is increased somewhat by an increase in
either or both components of the damping; (c) the ratio of arcs is at first
greater and then less than the average value; (d) successive differences
for the combined damping are at first greater and then less than for the
first power damping alone; (e) the influence of the second power damping
rapidly diminishes and the decrement approaches its value for pure
first power resistance. This is to be expected since a decrease in the
velocity to one half its original value reduces the second power damping
to one fourth its initial effectiveness.

2. Forced oscillations may be produced by twisting the torsion wire
periodically. To accomplish this, an arm is rigidly attached to the torsion
wire head and is driven back and forth harmonically by means of a con-
necting rod, slotted cross head and reduction gears. (See Fig. 5.) If
higher harmonics are desired in the forcing term, a combination of gears
such as are used in a curve-tracer may be employed to turn the torsion
wire. Following Routh's double approximation method a solution may
be developed for systems in which the amount of damping is small.

3. The influence of a constant damping factor, independent of the
speed, on the motion of an oscillating system becomes of importance in
several interesting connections. Thus from the equations derived by
Durand* for the surge-chamber problem in hydraulics, if the effect of
governor control is neglected and the motion of the water is expressed
solely in terms of displacement in the surge tower, there results ah
equation of the type

> E. J. Routh, Adanced Rigid Dynamics, p. 251, 1905. The solution given by Routh
(Art. 364) is limited to displacements so small that their squares and higher powers may be
neglected. It is possible, however, following Routh's general method to develop a solution
not limited by the magnitude of the displacement.

> W. F. Durand, op. cit., p. 321, 1912.

430 /. PARKER VAN ZANDT. [^Sm

Here there are present three types of damping: that varying with the
first power of the speed; that varying with the second power, and a
constant resistance independent of the speed. The model which has
been developed may be adapted to exhibit just such a type of resisted
motion.

In connection with electrical instruments subject to pivot friction and
in certain oscillating systems, Blondel and Carbenay^ have shown that
the oscillations are represented by an equation of the type

d^ idO\

i^ + B\J^) + Ce + D^o (or = ^ sin («/)). (21)

An electrical analogy is found in an oscillating circuit of radio frequency
in which spark resistance predominates.* An experimental study of
motion corresponding to the type equation (21) may readily be made.
It is beyond the limits of this paper, however, to enter into a discussion
of these allied problems in resisted oscillations. Enough has been sug-
gested, perhaps, to indicate the flexibility of the model developed and
the wide range of problems for which there is now a means of experimental

study.

Summary.

I. A complete solution has been developed for the equation

d^ ..ide\^

d^ idev

subject only to one limitation, namely:

(I ± 2iV^o)€'**''*

+ 1.

2. The solution has been verified by comparison with the results of
graphical integration and by experiment.

3. The oscillations have been shown analytically and experimentally
to be practically isochronous.

4. A solution has been developed by which the numerical values of
the successive turning points may be computed.

5. A model has been constructed suitable for a wide range of systems
by means of which it is possible to study oscillations damped by resistance
proportional to the square of the velocity.

> A. Blondel and F. Carbenay, La Lumidre £lec.. Nov. 27, Dec. 4, 11, 19 15, and July 29.
X916.

« A. W. Stone, Inst. Radio. Eng.. Proc. 2, pp. 307-427, Dec.. 1914.

nS"^] oscillating systems. 431

6. A method Is given for the determination of the coefficient of damping
in terms of the values of the successive turning points.

7. The model developed may be extended to exhibit: (a) forced oscil-
lations; (6) systems resisted by both first and second power damping;
(c) systems resisted by a constant factor, alone or in combination with
other forms of damping.

8. The prevalence of damping proportional to the square of the speed
in problems of resisted motion has been demonstrated by a survey of the
literature. A bibliography is given for the more important engineering
cases in which this type of oscillation occurs.

In conclusion, the author takes pleasure in expressing his thanks to
Prof. W. J. Raymond, Prof. B. M. Woods, Prof. F. E. Pernot and others,
whose valuable suggestions and friendly criticisms have been most
helpful.

Physical Laboratory, Univbrsity op California,
May, 191 7.

rSlOOND

432 ALBERT C. CREHORE. l&SS

THEORY OF CRYSTAL STRUCTURE, WITH APPLICATION
TO TWENTY CRYSTALS BELONGING TO THE CUBIC

OR ISOMETRIC SYSTEM.

By Albbrt C. Crbhorb.

IN a former communication^ an expression for the mechanical force
between any two atoms in their most general positions was derived
from the early form of electromagnetic equations proposed by Thomson.*
The assumption was made that these equations apply to the individual
electrons in the atoms, each revolving in circular orbits around a common
center determined by the positive charge, and that the atoms are neutral,
the total positive charge being numerically equal to the sum of the
negative charges. The total force between the atoms is obtained by the
summation of the forces between their various parts. To obtain the
forces between the two positive charges, and between the positive charge
of the one and an electron in the other, presents little difficulty because
the ordinary electrostatic forces apply in these cases. The problem then
resolves itself into that of finding the average force between two electrons
supposed to be revolving in circular orbits with uniform velocity, the
circles being in their most general positions in space. The result for
two atoms is given by equations (23)-(25), page 755 of the paper referred
to.

More recently* the same problem has been solved for two atoms using
the Lorentz form of the electromagnetic equations, involving the con-
ception of retarded potentials, which does not form a part of the older
Thomson equations. The expression for the average force is finally
developed in both cases in the form of infinite series of the inverse powers
of r, the distance between the centers of the two atoms. In the Lorentz
form the series begins with the inverse first power of r and all inverse
powers of r are present. When, however, the force is resolved along the
line joining the centers of the two atoms this series begins with the inverse
square term. If the distance between the atoms is large, the first term
is the only one which is effective, and we have in this result the suggestion
that this force may be identical with that of gravitation. A critical

» Phil. Mag.. June. 1915, p. TSO.

*J. J. Thomson. Phil. Mag.. April. 1881. p. 229.

*Phys. Rev., June. 1917.

^^] THEORY OP CRYSTAL STRUCTURE. 433

examination has shown that this force resembles that of gravitation in
many ways, in being always an attraction and never a repulsion, and in
being independent of the orientation of the axes of rotation of the atoms,
so that two crystals have the same pull no matter how they are oriented.
But, in the matter of the magnitude of the force, the theory demands a
force more than lo*^ times greater than the force of gravitation really is.
This result compels the belief that these Lorentz equations in their
present form without modification do not apply to the electrons in the
atoms of matter.

However, it is found that, when the expression for the inverse square
terms thus derived is multiplied by a factor proportional to the kinetic
energy of the electron itself, the correct value of the gravitational pull
is obtained. This has given some reason to hope that a modification in
the present form of the electromagnetic theory will be found that will
make it strictly applicable to the electrons in the atoms when in
their steady state not radiating energy. At any rate, I have
taken the liberty of introducing the factor demanded by the inverse
square terms into the Lorentz forms, thus making a modified form of the
present statement of these equations, and find that in so doing the average
force between two atoms, at the distances apart concerned in crystals,
agrees very closely with the result derived from the older Thomson form
of the electromagnetic equations. The first, second, and third terms of
the series become ineffective at this range, and, when account is taken
of the space-lattice formation of cubic crystals, the series begins with
the inverse fourth power, the even powers only following this.

This is precisely the form of equation obtained from the Thomson
theory, above referred to, the only difference between the two forms, after
introducing the modification, being a factor of 2 in some of the terms in
f"*. This factor of 2, however, makes some difference in applying these
results to crystals, and, of the two forms, the Thomson equations give
consistent results, showing stable equilibrium when we consider the odd
planes, as in rock salt, or the even planes as in copper; whereas, the other
form gives equilibrium for the odd planes only, but not for the even.
The consideration of crystals affords a test between the two forms, and
indicates that the Thomson form is to be preferred for these small
distances because the results using the odd planes, as in NaCl, KCl,
KBr, etc., fit very closely those using the even planes, as in Cu, Fe, Ag,
Pd, etc., whereas the other form does not.

The above remarks give some justification for the use of the following
equation for the force between two atoms at close range, based upon the
Thomson theory. Since the z or k component of the force, that acting

434 ALBERT C, CREHORB.

along the direction of the axis of rotation of the atom, is the only one that
we shall require, this is repeated here. The reason that the i and j
components are not required is that cubic crystals are so arranged that
these components each cancel out in summing the effects of surrounding
atoms upon a single selected atom. That is to say, the total force due
to all other atoms added together upon the one has the direction of its
axis of rotation.

Fm = e'{S(a/3)2(a/3)[+l.5«cosa+.75isina— 3.75m%cosa— 3.75Zn*sina

— 375^« cos a]i^^

— 2a* Sa*[+5.625n+3.75n sin* a+7.5^ sin a cos a+i.Sysn cos* a

— i3.i25(+Pn+4W*»+n' sin* a+^Pn cos* a

+4/n* sin a cos a+Pn sin* a+/' sin a cos a+mhi sin* a

+/m* sin a cos a) (i)

+59.o625(+Pn* sin* a+Pmhi+l^ cos* a+mhi* sin*a+m*n

+/*m*» cos* a+2Pn* sin a cos a+2lfn^n* sin a cos a)]r^}k.

In this equation e is the charge of the electron, a the radius of its orbit,
and P the ratio of its velocity to that of light. P is the number of elec-
trons in the first, and P' in the second atom. The summations indicated
in the coefficients are to be extended to each electron in each atom
respectively. /, m, and n are the direction cosines of the center of the
second atom with respect to the center of the first, referred to rectangular
axes, xi, yj, and zk^ having the origin at the center of the first atom.
The positive direction of the zk axis is that of the axis of rotation of the
first atom, so that the rotation is clockwise viewed from the positive end.
The positive direction of the yj axis is then defined by the vector k X k',
k' being the unit vector in the direction of the axis of rotation of the second
atom. This vector takes the direction of the intersection of the equa-
torial planes of the two atoms, being perpendicular to both k and Jfe',
and the positive direction is such that rotation from k to k' appears coun-
terclockwise viewed from the positive end. The quantities in the brackets
are functions of the position of the second atom and the direction of its
axis with respect to the first atom. These expressions become numerics
as soon as the position of the second atom is known, that is, when the
form of the space-lattice of a crystal is specified. The equation may be
written^

> The letter / has inadvertently been employed in two senses here, first as a direction
cosine in the functions, /i and /i, and then as the cube edge in f* and lr\ but no confusion
will arise from this.

J2J~j*-] THEORY OF CRYSTAL STRUCTURE. 435

F, = i?[U{h m, n, a) S (a/3) S (ap)/-« +/.(^ m, «, a) Sa* 2 a*/"* •••}*. (2)

The two quantities, Xp{aff) and 2pa*, are characteristic properties of the
atoms, and it is the purpose of this investigation to learn something about
them, as far as anything can be ascertained from a study of crystals. In
the case of the diamond^ an example has been given of the arrangement
of the planes and directions of the axes of rotation of each atom, to which
reference is made, in which it is found that the equation for the total
force on one atom due to the others may be written

F. = ^{ - 15.7225 2 (a/3) 2 (a/3)/-^ + 252.83 2a« ^aH-^ •..}*, (3)

where / is the edge of the elementary tetrahedron. Since this is ^2
times the cube edge, the above becomes

Fb = e«{- 3.93062(a/3)2(a/3)/-* + 3i.6o42a«2a«/-^ ...}*, (4)

where / is now the cube edge. When this force is equated to zero for the
equilibrium condition, we obtain, putting d = 31.604/3.9306 = 8.04,

2(ag)2(ag)^rf^8.04

p p*

A similar process has been carried out for the 124 atoms surrounding
the central atom in a simple cubic lattice of edge 4/, (see appendix) giving,
for the odd planes on one half of the cube only,

F, = e«{o.6oi3 2 (a/3) S (a/3)/-* - 4.6455 2 a« 2 a«/-^ •••)*, (6)

p p* p p*

and, for the even planes on one side only,

/?, = «»{- 0.49517 1 (a/S) S (a/S)/-« + 0.94059 1 a* 5 a*/-* •••}*. (7)

P " P P'

Equating these to zero gives for the odd planes

2 (a/3) 2 (a/3) „ ^ ^.

2a«2a» P P ' ^^^

p p*

and, for the even planes,

|(^g)S(flg)_^^i.90

2a*2a« P P ' ^^^

p p*

Experimental evidence from measurements with the X-ray spectrom-
eter gives the cube edge, /, and the form of lattice, but nothing else in
these equations. From such data we obtain merely the ratios in the left

» Phil. Mag., Aug., 1915, p. 257.

436 ALBERT C, CREHORE. [

hand members of (3), (8) and (9), the right members becoming simple
numbers when the values of / are substituted.

It is at this point where some kind of hypothesis concerning the quan-
tities a and j8, as relating to the atoms, is required. Heretofore, in the
papers referred to, the equal moment of momentum hypothesis was
introduced for lack of something better, though there seemed to be no
very urgent reason why this should be true. The quantity aj8 is pro-
portional to this angular moment of momentum, and, if this has the
same value for each electron in the atoms, the quantity in the numerator
of the left member of these equations is much simplified, becoming PP'
times a constant, these being the number of electrons respectively in the
atoms. If we should make this assumption, the values of Za' for the
different atoms in the crystals considered might be found in terms of this
constant.

Since, however, we have been able to calculate the speed of rings of
electrons, that of four being fi = .00846, and of eight j8 = .012, it is
evident that the natural tendency of the outer rings is to have a greater
actual velocity than the inner rings, if the outer ring has a greater number
of electrons. The equal angular moment of momentum hypothesis,
however, demands a greater velocity for the smaller ring, the velocity
being twice as great at half the radius, and the differential angular velocity
between different rings is very pronounced. On the other hand, the cal-
culation of the speed of rings seems to make the angular velocities more
nearly equal for different radii. If, for example, the radii were in the
ratio of the above speeds, that of the ring of eight being 12 and of the ring
of four 8.46, the angular velocities of the two rings would be the same.

It seems neither safe to assume an equal angular moment of momentum
nor an equal angular velocity for each electron in the light of present
knowledge. There is another investigation, however, which has a direct
bearing upon this matter. The tentative formula for the weight of an
atom due to the earth's attraction, to which reference has been made, is

where

F ^ZT- woei«2Sj8»Sj8«f^-* = *Sj8«, (10)

O A P E \P

the first summation being extended over a single atom, and the second
over all the atoms composing the earth, which is, of course, a constant
quantity if we are merely comparing the weights of two atoms, r^ is the
radius of the earth. This result indicates that the weight of an atom is

^^^] THEORY OP CRYSTAL STRUCTURE, 437

Strictly proportional to the sum of the squares of the linear velocities of
the electrons within the atom. Since the weights of atoms are constant
in the same locality, we may better assume that Xpff^ is constant for a
given atom. This quantity, however, does not occur, directly anyway,
in the equations (5), (8) and (9), by which we have to investigate crystals.
The solution for the speed of rings of electrons also shows that the
linear velocity is independent of the radius of the ring, and, hence, coupling
this with the suggestion as to the cause of the weight, we see that it is
quite possible that the radii of the orbits of the electrons in an atom may
change without affecting the speeds, or consequently, their weights.
But, as the radius increases, the angular velocity decreases proportionally
in order to maintain the same actual velocity. We may write the three
quantities for the atoms, with which we are concerned as follows,

Sa* = ai* + at* + az^ + • • • ap\ (12)

S(aj8) = - (ai*wi + ch?<at + aa^w, + • • • ap^cap), (13)

* c

Sj8* = -: (ai*«i* + cs^wt* + aj^wg* + • • • CLp^oi/). (14)

p c*

For a single ring atom the values of a and o) are the same for each
electron, and, for a ring of P electrons, we have

(15)
(16)

(17)

From these we have, for a single ring atom.

For a multiple ring atom we may use a mean square radius, mean velocity,
and mean square velocity, giving instead of (i2)-(i4)

(19)
(20)

So* '

'Pa*,

S(«l3)

P ,

S|8* =

■■ ^ c*«*.
c*

So*

= Pc«*.

S(o/3)

P -

= — «iOo*.

S^ =

' -J «»*Oo'.

(21)

438 ALBERT C. CREHORR,

The subscripts are used in connection with the angular velocities because
the two mean velocities obtained in this way are not exactly the same.
To show by how much they differ from each other in a special case, an
example is given below. We obtain from (i9)-(2i)

^ = ^ (22)

(23)

S(aiS) c<ai '
p

In a two-ring atom, let the radius of the outside ring be a, speed such
that fi = .012, and number of electrons eight. For the inside ring let
Oi « .402 a, Pi =» .00846, and the number of electrons four. Then, for
the atom

Sa* = 8a* + 4(.402a)* = 8.6464a* = Pa^* = i2ao*, (24)

Ha0) = 8a X .012 + 4 X .402a X .00846 = .1096a
p

= — witto* = — uiOf?, (25)

w C

P 12

Sj8* = 8 X .012* + 4 X .00846* = .001438 = -jwtW = -J ««*ao*, (26)

%(afi) _ .01268 6)1,

(27)

whence

Sj8* _ .01312 ^ (a£
p

(28)

OI312 w«

«J .^68=^°34. and -=1.017. (^9)

According to this example we do not make much error in assuming
that equation (i8), which is strictly true for the single ring atom, is also
approximately true for more complicated forms. Making this approxi-
mation, (5) becomes

Sa*|a*=|s/3»|/S«; (30)

with similar expressions for (8) and (9), the differences being in the values
of the constants.

But, by (10), we may consider that 2)p/3* is constant for a given atom,
and thus may find its value from the atomic weight. If W is the atomic

}SJ'^*] theory of crystal structure. 439

weight in dynes, and A the atomic weight referred to oxygen as i6,
then W is equal to A times the weight of the hydrogen atom divided by
1.008. Taking the mass of the hydrogen atom as 1.64 X lO"*^ grams,
and its weight 1.64 X 981 X lO"** dynes, the weight of any atom is

W » 15.96 X lO-^Ap = *Sj8», by (10). (31)

Hence

Sj8* - 15.96 X lO-^Ap/k » kiAp, (32)

where

ki = 15.96 X io-«/Jfc. (33)

Putting this in (30) gives as the equilibrium condition for the diamond
form of lattice,

Sa«Sa* - ^A^^*. (34)

For a crystal like rock salt we obtain from (8) a similar form to this,
with the constant^ d equal to 7.73 instead of 8.04, being obtained from the
odd planes of the simple cubic lattice, and for copper from (9) a similar
form with the constant 1.90, obtained from the even planes of the simple
cubic lattice, in this case the even planes being the only ones present.

These formulae have been applied to twenty different crystals belonging
to the cubic or isometric system, some of which are known to have been
examined by the spectrometer and others not. In certain cases the
formulae give Xa* directly, when there is but one kind of atom in the
crystal, as diamond, copper, silver, iron, etc., but in most cases they
give products Xg/i^XpfG* for the two kinds of atoms that enter the crystal.
The separate values in this case for each kind of atom may only be
obtained when one of them is known from some other crystal.

The remarkable result is obtained that Xa* shows a gradual progression
from element to element, the irregularities being of about the same order
as the irregularities in atomic weights. The curves obtained from these
twenty crystals, containing as many separate kinds of atoms altogether,
are shown in Figs, i and 2. The best values of 2a* and S(aj8) to agree
with the different crystals are plotted for each of the twenty elements
as separate points.

It is found, however, that, in order to make the points fall near the
curves, the calculation must be made for some of the crystals as though
there were double instead of single atoms at each point of the space
lattice. The points either fall near the curve for the correct form of
lattice and proper number of atoms, or very far from it for a wrong

^ See equation (5) above.

4ZZ ALBERT C CMEBOME.

rrnraTiTn- Tbe oocapoitods of solpfaur may serve as an Ohistratioii.
Tiie crysiak, 2iiK±^eDde ,ZnS, diazDOod lattice, and galena ^PbS, rock
sfth larDce . havie been stDdkd by the spectrocDeter, and the measured
fr>ar;T.g^ z£ t^ pCanes agree with those calculated from their densities,
or tbe asEvizipoco that there is but a single atom of zinc and a stng^
izzin :c fc'rrc-iir at the pocnts of the lattice. The points, as ralnibtfd
ibr^^ f ccimilae with this spacing, do not fa3 anywhere near the
xn. if we had calculated the lattice cq the assumption that the
£r:c2^ DC eiftc^ were dccib-je at the various pccntss we should have obtained
ML ec|:t f :c zbe eaanectary cube 2- • gicaua than before, and the product

d have been 2 * » greater, depoiing upoc the fourth power of
3 for TJnc and sulphur thus obtained fall near these curves.

Table L

V«w s si^ Cm ■■ Fig. t U

X»C: Rock s»i C

Kc: - o

k: - n*

AfO - S

.\5:Bc - C

P^i^ ~ K

F*rTe - Ca

:. CiO * Fe

-I ZzS ' 2m.

-'* CaFi P-jorsotr Br

:^ Fe>, :--ir --r» Pi

-- C* C-vroer Ag

-t F« * Tc

M - Pi

?: 'Ax

—^ "itii iiiise ir A.i*t;!iii "v^ ^;iLi\'e tc iss-rzsf r^r-i-r Jtrrcas cc ttMZ. anc tocr ol
Lirnnr- zn^r^iaii it :o* it tbe rcilrts c£ tbe lirrlce. r-.rV^g I greater by
• * i^^i ~' i-^ 2* • tr^iJi r: rexf v 25 rr tmcj tbe rvx::t5 fiJZ aesir the c
ir* riii ipi-rrrrnitcsr r.ig ieiritifly sifct'jei tb? c'jesci:^ of tin
It rre tit-m^fnTT r^re -c tr<e s:\xi:>e lirr.:^, ^-e cxz dx a&soose that

If^^'] THEORY OP CRYSTAL STRUCTURE. 44 1

there are double atoms in zincblende and quadruple atoms in galena at
the points of the lattice, and, according to the hypothesis that 2j8* is
proportional to the weight, we are led to believe that 2a* is not always
the same in some kinds of atoms at least.

If we consider that manganblende, MnS, is a simple cubic lattice like
rock salt, the values of Sa* fall near the curve by assuming double atoms.

Another crystal containing sulphur is iron pyrites, which has been
described by W. H. and W. L. Bragg in " X-rays and Crystal Structure "
in much detail. If we apply the same equilibrium formula to this as
applies to diamond, zincblende and fluorspar, we obtain points close to
the curve by assuming double atoms. How it is possible that the same
equilibrium condition may be applied to such different lattices is shown
in the appendix hereto.

The point we are now making is that these four crystals fall into line
if we admit that there are two different kinds of sulphur atoms, the
same kind of sulphur in ZnS, MnS, and FeSs, and a different kind in
PbS, their weights being the same in each case, but 2)a*, S(aj8) and «i
differing for the two kinds. To fall upon the curve in Fig. i, the sulphur
atom should have approximately

Sa* = 300 X lo-^'— .

P V

In ZnS, MnS, and FeSj the value required is (J)*'* =* .397 of this, and in
PbS (J)*^* = -1575 of it. The value corresponding to the curve does not
occur in these crystals, but it seems likely that it does occur sometimes,
perhaps in other crystals.

A similar statement may be made for the atoms of oxygen, chlorine,
and bromine. By admitting two possible forms for these three atoms,
having the same atomic weight in each case, all of the twenty crystals
considered are brought into line. Chlorine and bromine in the halogens,
NaCl, KCI, KBr have values of 2)a* corresponding to the curve, but, in
AgCl and AgBr, the CI and Br have values (i)*'» = .397 of that in KCI
and KBr. In the mineral melaconite, CuO, the oxygen has a value of
2)a* (i)*'* of that given by the curve. This is the only crystal considered
containing oxygen. It seems likely that a value of oxygen will be
obtained from other crystals which agrees with the curve. The only
crystal containing fluorine, CaF2, gives a value close to the curve. We
seem justified in expecting to find in other crystals a value equal to
(i)*/» of this for F.

In Fig. 2 the atomic weights of the elements are plotted with reference
to the atomic numbers, and a mean curve drawn through the points.
By equation (32) we may obtain from this the values of 2/3* when the

442

ALBERT C, CREHORE.

abscissae of this curve are multiplied by the constant k\. The approximate
value of S(c/S) may then be derived by multiplying together the corre-
sponding abscissae of Za' in Fig. i and SjS*, and taking the square root of
the product, according to (i8). The curve of X(afi), thus obtained, is
shown in II., Fig. 2. And, according to (22), we may obtain the mean
angular velodty of revolution of all the electrons in the atoms by dividing
X{aP) by 2)a*, which is shown in Curve III., Fig. 2.

9.(\

" AC\

1 40

ni
/•

■

fcU

'/x

ifl

XX)

•

aooo

Curves I., II. and III.

Curves I.. II. and III.

Fig. 1.
r2a«Xio". Scaled.

R\P

Z a* X 10^ sq. cm. Absolute scale B,
p

The character of the curve of 2a*, Fig. i, supports the theory that the
electrons are distributed in rings resembling those originally calculated
by Thomson, and that the volume of the sphere enclosing the rings
increases by uniform steps as electrons are added. The volume of a
sphere enclosing the orbits of the electrons, on the assumption that they
are arranged in a plane, may be taken roughly proportional to the cube
of the mean radius. The mean square radius is proportional to Xa*/N,
where N is the atomic number, and the volume, therefore, proportional
to (2a*/iV)*'*. If the volume is also proportional to the number of
electrons, and this again to the atomic number, we derive the equation

I -TT- 1 = iV times a constant.

Hence

2)a* = 6iV*'*, where 6 is a constant. (35)

The values of 2a* read from the curve, and iV^* calculated from the
corresponding atomic numbers, are as follows:

Vex. XI
Has. J

THEORY OP CRYSTAL STRUCTURE.

443

Table II.

AT.

ATI.

S4I*.

2a*

46.3
147.3
289.4
469.
679.
920.

1.189.

1,485.

1,806.

110
442
830
1,263
1,745
2,235
2,713
3,213
3,712

2.38

3.00

2.87

2.7

2.57

2.43

2.28

2.16

2.06

The constancy of this ratio in the above table is as nearly perfect as
we should expect, were it exactly true that the volume of a sphere en-
closing the outside rings increases by equal steps for the addition of each
electron, because of the approximation we are forced to use in deriving
the result in (35).

The curve for X(a$) in Fig. 2 is very nearly a straight line for the atomic
numbers above 40, Zirconium; but the line does not pass through the
origin. If we dvide S(ajS) by N, to obtain the average moment of momen-
tum per electron, these exhibit a gradual increase with increasing atomic
numbers. The average for the heavier elements is sufficiently constant
to have suggested the idea that the angular moment of momentum for
each electron is constant,^ but, in the lighter elements, these values vary
considerably from those in the heavy elements.

The curve for the mean angular velocity of each atom. III., Fig. 2,
shows a decrease in the frequency for an increase in the atomic number.
The frequency becomes very large for the lightest elements. For
carbon this frequency, as we shall see by the two examples, is about
.78 X lo^*. The average frequencies for the other elements are obtained
from the ratio of the abscissae of Curve I. to those of Curve II., Fig. 2,
and, since Curve II. approaches zero much more rapidly than Curve I.,
the ratio and the frequency become rapidly larger for decreasing atomic
numbers. The fundamental value for hydrogen has been found else-
where* to be of the order of 10^, considerably greater than that for carbon.

If one identifies the frequencies of revolution of the electrons in their
orbits with optical frequencies according to Bohr's theory, the radii of
the orbits, being inversely as the frequencies, come out much greater
than the orbits indicated by the above average values of the frequencies.

^ Moaeley, Nature, Jan. 15, 1914, and F. A. Lindemann, Nature, Jan. i, Feb. 5, I9I4*
> Unpublished.

444

ALBERT C. CREHORE.

It is possible, of course, that a small number of electrons circulate in
large orbits with smaller frequencies, and that the rest have very much
smaller orbits and higher frequencies, so far as we can tell from these
average values. If this is true, then the electromagnetic forces, as de-
veloped in infinite series cannot be applied to these outside electrons,
because the force-series becomes non-convergent when the distance
between the centers is comparable with the diameter.

[0 So m

200 400 600

Fig. 2.

1000 c

Curve I. r2/8«. Scale i4.
kiP

<i
Curves II.. IV. and V. —S {afi) X io«. Seale C.

k\ p

Curve I. 2/8«Xio«.
p

Absolute scale B,

Curves II.. IV. and V. 2 {afi) X lo". Absolute scale D,

Curve III. — X lo"*. Absolute scale F.
c

There have been cogent reasons for believing that a few so-called
** valency " electrons do have these larger orbits, and, if so, electro-
magnetic theory should still be capable of dealing with the matter as
long as they are in the steady state not radiating energy. A more com-
prehensive method of analysis is required, however, before any equations
embracing these outside electrons can be obtained. It seems to be
necessary to show that such electrons may be permitted by the theory

Na*^*] THEORY OP CRYSTAL STRUCTURE, 445

without disturbing the equilibrium, on account of their mutual inter-
ference and the resulting perturbations produced. The evidence in
favor of their existence is greatly strengthened by the very recent work
of J. Frenkel^ in calculating on Bohr's theory the '* intrinsic potentials "
of, and the Volta contact electromotive forces between, metals and non-
metals by means of the supposed existence of a small number of electrons
having these large orbits. In the way it is done, it is these large orbits
alone that are responsible for the effects, the small ones being of no
avail. He also calculates the electrical energy of the quasi-surface con-
denser thus produced, making it the same as the well-known energy of
surface tension, which is thus explained in terms of atomic structure.

It is noteworthy that the order of magnitude of the frequencies in
Curve III., 10^*, is the same as that of characteristic X-ray frequencies.
It has been pointed out before that these frequencies of revolution should
not be related directly to these X-ray frequencies. The one may be a
function of the other, but the evidence goes to show that the X-ray fre-
quencies are functions primarily of the atomic number and a series of
ordinals alone, which would give the smooth character to the Moseley
curves. The dependence upon Xa^, 2(aj8) and rotation frequency
secondarily may account for the small curvature observed in his curves.
The large number of lines in the X-ray spectrum is alone almost sufficient
to make this independence of the two kinds of frequencies probable.

The Absolute Values of the Constants.*

The curves in Figs. I and 2 have abscissae which are dependent upon
the absolute value of the constant ku defined by the equations (33) and
(11) above. It is necessary to know the value of this constant before
Xa^, etc., can be found in absolute measure. It is possible to find ki if
we know the number of electrons in any one atom, their arrangement in
rings, and their speeds. We shall make a tentative assumption as to the
carbon atom, and derive from it the value of ki. The reason for giving
the curves in terms of this constant is so that, if any one prefers a different
assumption as to carbon or any other atom, the absolute values may be
more readily obtained. Let us take 20* for carbon as given in the ex-
ample, equation (26). Then, by (32)

* J. Frenkel, On the Surface Electric Double-layer of Solid and Liquid Bodies, Phil. Mag.,
April, 191 7. p. 297.

' The following numerical estimates of absolute values must, of course, be considered as
tentative and subject to revision. They are chiefly based upon the calculation of actual
velocities of electrons in rings according to electromagnetic theory. The process of making
such calculations is long and tedious and should be checked both as to method and errors of
a mechanical nature. It should be emphasized that the methods outlined here should yield
the proper numerical values when these velocities of electrons in rings are accurately known.

446 ALBERT C. CREHORE. [j

kx = .001438/12 = 1. 198 X I0-*. (36)

The value of v, as in (8), is 7.73. Hence, to convert the abscissae of
the curves into absolute measure, multiply those of
Curve I., Fig. i, for 2a* by k\ X lO-^/v « 0.155 X lO"**,
Curve I., Fig. 2, for ZjS* by *i = 1.198 X lO"*,

Curve II., Fig. 2, for 2)(a/8) by -7= lO"* = 0.431 X lO"",

Curve III., Fig. 2, for ^y = ^f^ = ^ by ^i X io» - 2.78 X io».

For the carbon atom the reading from the curve for 2a* is 34.95.
Multiplying by the factor 0.155 X lO"* gives in absolute measure
2a* = 5.42 X lO"*® sq. cm. Equating this to (24) gives the radius of the
outside ring a = 0.792 X lO"*® cm. This absolute value for the radius
is in accord with former results, all of which show that the radius is a
very small quantity compared with the distance between adjacent
atoms. The edge of the tetrahedron in diamond is 2.528 X io~* cm.,
319 times the radius above determined.

We obtain also the mean angular velocity and frequency of revolution
for carbon from (25) to be w = 4.80 X 10", and n = 0.764 X lo^*.

Had we made a different assumption for the carbon atom, the order
of magnitude of these quantities is not greatly changed. For example,
let the carbon atom be supposed to consist of a single ring of six electrons,
for which j8 falls between .00846 and .012, say at .010. The exact value
of j3 for a ring of six has not been calculated at this writing. From this
assumption 2/3* = 6 X lO"*, and

ki « 0.5 X 10-*. (37)

Hence (*i X io-")/v = 6.47 X lO"**, and 2a* = 6.47 X 3495 X lO"**
= 2.26 X 10""*® sq. cm., and the radius of the ring a = .614 X lO""^
cm., instead of .792 according to the former assumption. Also
2(aj8) = 6aj8 = 3.684 X lO"", and

X^ 2(a^) 6X10- 3.684X10- ,,3^,^^«. (33)

2(aiS) 2a* 3.684 X lo"" 2.26 X io-«> "^ c

Hence the angular velocity o) = 4.89 X lo^*, about the same as the
mean value of o) in the former example.

Speculation as to the Kind of Atoms in the Interior of the Earth.

By means of the absolute values of k\ in (36) and (37), obtained from
different assumptions as to the structure of the carbon atom, we are
enabled to get an approximate value of the constant k in the weight

1%:^] THEORY OP CRYSTAL STRUCTURE. 447

equation (10). For, by (33) k = 15.96 X iQ-^/ki. For the first kind

of carbon atom we have

k - 13.32 X 10-", (39)

and for the second

* = 31.9 X 10-". (40)

A knowledge of this constant enables us to find from (11) 2^/3* for
the earth, that is, the sum of the squares of j3 for every electron in the
earth. This is

S/3« = — ^, (41)

m^ '

where r^, the radius of the earth may be taken as 6.367 X lo* cm., and
Wo, the mass of the electron, as .898 X lO"*' grams, and t * 4.77 X lO"*®
electrostatic units. With the first kind of carbon atom we get

|/3* = 21.12 X 10^ (42)

and with the second

|/3« = 50.56 X 10^. (43)

Dividing this by the total number of electrons in the earth, we obtain
a value of j3 for the average electron in the earth. The volume of the
earth is 1.083 X 10*^ c.c, mean density 5.5247 db .0013, and mass,
therefore, 5.984 X 10*^ grams.

If we take the number of electrons per atom as proportional to the
atomic number, then the number of electrons per gram of any substance
is constant. This may be shown as follows. It is well known that a
cubic cm. of a perfect gas, under the standard conditions of temperature
and pressure, contains the same number of molecules, say N. If d is
the density of the gas, then the number of molecules per gram is iV/d,
since i Id is the volume of a gram of the gas. If M is the molecular weight,

M = n\A\ + n%At + n%A% + • • •,

where A\^ Ai, etc., are the atomic weights of the various atoms in the
molecule, and «i, n2, etc., the numbers of these atoms respectively. If
the number of electrons in the atom, P, is proportional to the atomic
number or approximately to the atomic weight, we have

A\ = hP\\ i4t = 6P2» etc.

Hence M = 6(niPi + «^P2 +•••)= *-?» where P is now the nimiber
of electrons in one molecule. It follows that the mass of all the molecules
in one c.c. of the gas is tn = vol. X density = d = hNM = JfeiVP,
where h and k are constants. Hence, for two different gases

did' = PIP'.

448 ALBERT C. CREHORE, [ISSS

Since the number of molecules per gram is N/d^ the electrons per gram
are proportional to NM/d and to NP/d. And, since N is constant, and
P/d = P'/d\ the electrons per gram are the same for different gases.

The number of electrons per gram is, therefore, the same for all forms
of matter, liquids and solids, because the atoms have the same weight in
any of these forms, and the number of electrons per atom may be supposed
to be the same under all conditions.

Knowing that the number of atoms of hydrogen per gram is 6.05 X 10^'
approximately, and considering that the hydrogen atom has but a single
electron, this number may be taken as the number of electrons per gram
for any substance.

Multiplying the electrons per gram by the mass of the earth in grams
gives the total nimiber of electrons in the earth as approximately

6.05 X io*» X 5.984 X io«^ = 36.2 X 10". (44)

The mean values of /3* and fi for the average electron in the earth may
now be found by dividing 2^/3* by the total number of electrons, giving,
for the twelve-electron carbon atom,

/3* = 0.5856 X io~* and fi = 0.00765.

G>mparing this value with p for a ring of four electrons, 0.00846, and
for a ring of eight, 0.012, shows that it is a little less than the value for
the ring of four. Comparing with the values of obtained for hydrogen
in its different conditions, namely, 0.00738, 0.00369 and 0.00246, cor-
responding to the first, second and third states of hydrogen respectively,
shows that it is about the same as the value in the first state of hydrogen.
The majority of the electrons in the atoms of the earth's crust with which
we are acquainted have, according to the Curve I., Fig. 2, a value of j8
considerably in excess of 0.00765, so that the electrons at the center of
the earth must have a value less than the mean. Such a value would be
too small to agree with any of the forms of atom except hydrogen. It
has been pointed out elsewhere that there is no evidence for the existence
of hydrogen in the first state, and that the normal condition for hydrogen
is the second state in which p = 0.00369, which is less than the mean
value for the earth. If the interior of the earth were composed of hydro-
gen, the density might still be very large, as this is due to the compact-
ness, or the interspace between atoms, rather than to the character of
the atoms themselves. We know that the density at the center must be
in excess of 5.52, the mean value, because the average density of the
surface, that of the earth's crust, is considerably less than the mean
density.

XS"s^*] THEORY OP CRYSTAL STRUCTURE. 449

The above is at least an interesting speculation indicating that we
have obtained a method that may eventually yield some information as
to the nature of the large body of atoms in the interior of the earth. Had
we carried through the calculation based upon the six-electron atom the
mean value of P would have been about half again larger than that given.

Bulk MoDin-i.

In a former paper^ a formula was given by which the bulk modulus of a
crystal can be calculated from a knowledge of the forces interacting
between the atoms. This may be defined as the ratio of the pressure
per sq. cm. in grams weight to the change in volume per unit volume,
that is, the substance is more incompressible the greater its bulk modulus.
This formula is

^^J idF

9 X 981 ? dl

Table III. gives the values of M for several crystals as calculated by
this formula. The measured values of those that are known are given
in the third column, together with some crystals for which the modulus
has not been calculated, in order to show the great range of measured
values for different crystals, as well as to show that the order of magnitude
of the range of calculated values is the same.

The calculated values are larger in every instance where the value has
been measured, indicating a more incompressible substance than it is
measured to be. The range of values among those calculated is about
six to one from greatest to least, while the range of the measured values
is 66 to one. The measured value for tourmaline is well toward the top
of the list of calculated substances, showing the same order of magnitude.

Measurements of the bulk modulus must be subject to considerable
error because we have no means of compressing a substance except to
press upon it with another substance having a similar character. In such
a case it is difficult to say that the two substances do not interpenetrate
each other to a certain extent at the surface at least. By exerting pressure
by mercury upon a piece of steel, for example, Bridgeman has shown
that it is possible to force the mercury completely through the steel, so
that, when broken afterwards, it shows an amalgamated surface over the
entire break. If any interpenetration whatever occurs it will have the
effect of reducing the value of the bulk modulus obtained, which is in
line with the results shown in the table. Moreover, if the same substance
is used to produce the compression for a number of different substances,
the amount of such interpenetration should vary widely with the kind

* Loc. cit.

450

ALBERT C. CREHORB.

[

of space lattice of the crystal. Hence, great variations in measurements
should be anticipated if there occurs any interpenetration, which is
again in agreement with the results in the table.

Table III.

CrystAl.

Bulk Modulus.

Altoite, PbTe . .
Diamond, C . . . .
Galena, PbS . . . .
Fluorspar, CaF}.
KI

AgBr

KBr

Zincblende, ZnS. .
Iron Pyrites, FeSi,
AgCl

Calculated.

Meaaurad.

21,700 X10«

17,520 "

15,300 "

11,330 *•

860X10*

8,380 "

7,660 "

7,380 "

6,500 "

Rock Salt, NaCl . . . .

KCl

Melaconite, CuO . . . .
Manganblende, MnS.

Barite

Beryl

Quartz

Topaz

Tourmaline

5,470
4,400

4,380
4,000
3,660
3,570

II
II
II
II

906 "

246

138

535

1,384

387

1,694

9.140

If the same crystal were measured with compressing substances which
vary as widely as possible in their properties, it might prove to be the
case that different values of the bulk modulus would be obtained for the
same crystal, which would help to confirm the above suggestions.

Review and Summary.

I. The problem of finding the average mechanical force that one elec-
trical charge exerts upon a second charge, each being in uniform circular
motion, has been solved, both for the form of electromagnetic equations
originally proposed by Thomson, and for the more recent form of Lorentz
involving retarded potentials. It was shown in a former paper that the
Lorentz form without modification cannot be applicable to the electrons
in the atoms of matter, because their application would produce a force,
varying inversely as the square of the distance between two pieces of
matter at a great distance apart, more than lo*^ times greater than the
existing force of gravitation. When, however, the result thus obtained

5^/^] THEORY OP CRYSTAL STRUCTURE. 45 1

is multiplied by a factor proportional to the kinetic energy of the electron
itself, the attraction agrees very closely with that of gravitation. When
such a factor is introduced into the Lorentz equations the average force,
at the distances considered in crystals, reduces to very nearly the same
form whether the Lorentz or the original Thomson equations are used,
the same within a factor of 2 in some of the terms in r"^. This is regarded
as some justification for using the original Thomson equations for these
ranges of distance. The equation for the force between two atoms,
derived from these equations as given in a former paper, is here applied
to twenty crystals belonging to the cubic system.

2. Equilibrium conditions are derived for several forms of space
lattice. In each of them the only unknown quantities are 2)pa* and
2)p(a/3), summed for each electron in the atom concerned, a being the
radius of its orbit, and /3 its speed in terms of the velocity of light.

3. Some hypothesis is required before either the relative or the absolute
values of a and /3, the unknown quantities pertaining to the atoms, can
be found. In a former paper the equal moment of momentum hypo-
thesis for each and every electron in the atoms was adopted for the lack
of something better. This hypothesis led to certain serious difficulties,
in that it demanded that in such crystals there should be double atoms
instead of single atoms at each point of the space lattice. The evidence
of the spectrometer has made this view untenable.

4. The hypothesis as to the atoms adopted in this paper, instead of
the equal moment of momentum hypothesis, is that 2)p ^ is constant for
any given atom. A former work has indicated that the gravitational
attraction between bodies at a distance is proportional to the product of
the sum of the squares of the speeds of the electrons summed over each
body separately. Since the weight of an atom is constant, it is reasonable
to suppose in view of the above that the sum of the squares of the speeds
of the electrons within it is constant.

5. This hypothesis avoids the difficulties in which the equal moment of
momentum hypothesis involved us, by which 2p(a/3) is constant for a given
atom. According to the new ** 2p jP = a constant " hypothesis, the
same atom may take two or three diflferent forms without altering 2p jP.
This means that the moment of momentum is not constant, but may
have two or three diflferent forms corresponding to changes in the radii.

6. The values of ZpO^, 2p (a/3) and 2p /S* have been found for each of
twenty diflferent atoms that enter as many diflferent crystals, and are
plotted as curves in Figs, i and 2, against the atomic number in terms of a
constant multiplier. If an assumption is made as to some one form of
atom, these constants may be determined and the above values found for

452 ALBERT C. CREHORE.

any atom in absolute measure. The assumption that must be made con-
cerns the number of electrons in some atom and the speed of each ; but, for-
tunately, the radii of their orbits is not required. It is also fortimate that
the speed of electrons in rings has been previously determined numerically
for a ring of 4 and a ring of 8 electrons. These speeds come out inde-
pendent of the radius of the ring, and dependent only upon the number
in the ring. The carbon atom has been selected in making this assump-
tion, and two forms of it are given as examples, a 12 electron, and a 6
electron atom. The values of the constants do not differ greatly for the
two supposed forms of carbon. By means of this an absolute scale has
been determined in Figs, i and 2. It is there given for the twelve atom
assumption.

7. Two secondary curves, II. and III., Fig. i, are given as the alter-
native values of Xpa* for some atoms. These curves have abscissae
I /2^f* and i/2*'* of those in Curve I. respectively. The value of 2pa* for
the atoms O, S, CI and Br may occur in crystals in any one of two or three
forms, their weights remaining the same, but their radii changing. This
change in the radius does not affect Xp fi^. Not more than two of these
forms occur in the crystals considered in case of the above elements, but
it is predicted that the third form will appear in some crystals not yet
studied. For example, sulphur occurs in zincblende, manganblende and
iron pyrites with a value corresponding to Curve II., and in galena with
a value corresponding to Curve III., and no crystal gives a value cor-
responding to the principal Curve I. On the other hand, chlorine in
NaCl, KCl, and bromine on KBr give values on the Curve I., and in
AgCl and AgBr in Curve II., but no value in the crystals studied falk on
Curve III. for these elements.

8. Curve II., Fig. 2, with the secondary Curves IV. and V. give the
values of Xp (a/3), proportional to the total moment of momentum of the
atom. These three curves have abscissae in the ratios i : 2*'* : 2*'*
and they respresent the alternative values that an atom of sulphur, for
example, may possess in different circumstances. These curves are nearly
straight lines for atomic numbers above 40, and, were it not for the al-
ternative values in these three curves, would give good reason to suppose
that the moment of momentum for each electron is nearly constant.

9. Curve III., Fig. 2, gives the average value of w/c for each atom,
from which the average frequency of revolution may be obtained. Ac-
cording to it, the average frequency approaches a nearly constant mini-
miun for the heavier elements, but may be very large for the lightest
element, hydrogen.

10. A proof is given to show that the curve for Xp a* in Fig. i is in good

^^'] THEORY OP CRYSTAL STRUCTURE, 453

agreement with the theory that atoms may be formed in rings in a plane
as in the Meyer figures, the volume of the enclosing sphere increasing by
equal steps for the addition of each electron, but this offers no explanation
for the two alternative values for the same atom corresponding to Curves
II. and III.

1 1 . According to the twelve-electron-atom-assumption the radius of the
outside ring of eight is determined in centimeters to be 0.792 X io~*®.
This is I /319th part of the edge of the elementary tetrahedron in the
diamond. According to the six electron assumption it is 0.614 X lO"^
cm. The mean frequency of revolution is about .76 X lO*' in both
examples, that is, the order of magnitude of characteristic X-ray fre-
quencies.

12. A knowledge of absolute values leads to a determination of the
constant in the equation for the weight of an atom, from which Xg^ for all
the electrons in the earth is found to be 21.12 X lo**. Dividing this by
the total number of electrons in the earth, which is equal to the mass of
the earth in grams times the electrons per gram, a constant quantity,
namely 5.984 X 10^ X 6.05 X lo** = 36.2 X 10*®, the value of /P for
the average electron in the earth is .585 X io~^, and fi = 0.00765. This
result leads to a speculation that the interior of the earth may be hydro-
gen, or, at any rate, the very lightest of the known elements. That a
result, found in such a manner comes out within the range of the possible
values of /3 for any atoms helps to strengthen the theory that the gravita-
tional force is proportional to 2)j3*. It is, at least, an interesting specula-
tion 4>ecause it suggests for the first time a possibility of finding the kind
of elements that make up the interior of the earth and possibly other
heavenly bodies.

13. From the mechanical forces interacting between the atoms in a
crystal a formula for the bulk modulus was derived in a former paper,
which is here applied to several crystals. A comparison with measured
values in Table III., in the few cases where measurements are known,
shows that the crystals are invariably more incompressible according to
calculation than they are measured to be, although the order of mag-
nitude of the two results corresponds. It is suggested that the great
variation in the measured values among different crystals shows a certain
degree of interpenetration of the compressing substance and the substance
compressed. This would always have the effect of reducing the apparent
incompressibility, in the direction that the measurements indicate. New
measurements of these quantities may well be made, employing the same
crystal with different compressing substances. If different values are then
obtained, the fact may be attributed to different degrees of interpene-
tration. Liquids, however, which have no space lattice formation may

454 ALBERT C. CREBORE. SSE

act very much alike. If the compressing substance were a solid it would
be better.

14. The derivation of the force equations from the cubic space lattice
formation is given in an appendix 90 as to interfere less with the con-
tinuity of the argument. A section of this is devoted to a coosideratioo
of the error that is made in ne^ecting the more distant atoms in the
crystal than those included in a cube of edge four times the elementary
cube. This error is appreciable but not excessively great, as indicated
by calculating the total (or<% due to all atoms along selected radii in the
crystal to an infinite distance. The chief effect of such error is to alter
somewhat the values of the constants d, v and u in the equilibrium con-
ditions (5), (8) and (9) for different lattices. But, the fact that we obtain
good curves in Figs. I and 2 from different crystals and different lattices
goes to show that there is not a great error in these constants due to the
omission of atoms more distant than those calculated.

Vol. XI
Nas. J

THEORY OF CRYSTAL STRUCTURE,

455

Table IV.

No.

1
1
2
3
1

2
3
4
5
6

1
2
3
4
5

6
7
8
9
10

1
2
3
4
5

6
7
8
9
10

11
12
13
14
IS

3
4
5
6
7

4-1V6/

-1V5/

4-iV6/

II
II

II
II
II
II

II
II

4-V6/

4-iV6/

4-1 V6/

-}>fe

II
II
II

II
II

4-JV6/

-1-^5

4-1 V6/

II
II

• /•

-2V3/

2V3/

-|V2/

-|V3/

V6/

2V2/

-V3/

<3l

-V2/

V5/

+ "

__ II

4-"

__ II

+'•

-iV3/

V2/

4-V2/

V6/

__ II

4-"

__ II

4-"

__ II

2V3/

-iVi/

<3l

-V2/

Vs/

4-1 V6

-1

+1V3

II
II

II
II
II

II
II

+iV6

II
II

+ i>/3

II
II

4-iV6

-i

II
11

II
II

+ IV2

II
II

-\yl6

4-JV2

II
II

+AV30

II
II
II
II

II
II
II
II
II

4- "

0
II

II
II
II
II
II

+ iV3

_ II

+ "

__ II

II
II
II

II
II
II
II

4V3

-!V2

-IVS

-tVlO ,
4- "

1 __
iVlS

II
II
II

II
II

-J>S

II
II

-iV6

-1>I3

-iV2

II
II
II

II
II

-1

-IVJ

-i

II
II

-Hio -I'^Vis

109^ 28'.3

109^

109*

28'.3

109** 28'.3

Cos a.

Sin a.

+IV2
II

4-1
II

-i

+ Direction of Axis.

jr.

0» 12
0,8
0, 10

II
II
II

4-1

-i

4-1
II

II
II

II
II
II

+iV2

4-1

0. 12
0,8
0, 12
0,8
0. 10

0,8
0, 12
0, 10
0,8

0, 12
0, 10
0, 12
0,8
0, 10

0. 12
0,8
0, 10
0.7
0.9

0, 11
0, 10

0. 12

0,8

0, 12
jo, 8
'0, 10

4-1 V2 I "
0, 12

0, 14
0, 16
0, 18

0, 14
0, 16
0, 14
0, 16
0, 18

0, 16
0, 14
0, 18
0, 16

0, 14
0, 18
0, 14
0, 16
0, 18

0. 14
0, 16
0, 18
0, 15
0, 17

0. 13
0, 18

0, 4-6-1

0, 14

0, 16

0, 14
0, 16
0, 18

0,8

0. 14
0, 16

0, 10 0, 18
0, 12,0, 14
4-1 V2O. 8 0, 16

II
II
II
II

II
II
t*
t*
II

II
II
II
II
II

II
II
II
II
II
II
II

II
II
II
II
II

456

AISBMT C. CREBOKE.

• 8

lf«. lf«.

-1 8 +Ha H-^'S/

10 ^ " + "

11 '

• 12

^14

18
1
2
3

7
8
9

10
11
12
13
14

15
16
17
18

-JV& +2^*2/

it
It
II
II

«i

It
II

II
II
II
II

+ "

-V3/

+ "

II
II

+ '•

11
II

+ "
+2V2/

11
II
II
II

It
II

It
It
tt

II
II
II

-»>Dirwtte

mmiAMm.

«. CM«.

■ia«.

*•

0^ k), 16

-JV3/

^Isi

-hi^lo

+1^10

-AVr5109*28'J -1

+1^

0, +6-1

^ II

0.10,0, 18

-h "

-» ••

0. 12,0, 14

-h "

M£

-IV6

+1^7

-i^5

__ l«

0, 100. 18

-h "

0, 8 iO. 16

^ II

0, 12 0, 14

-h "

0, 10 0, 18

^ 11

0.8

0.16

V2/

- 1

+ "

0. 12 0, 14

^ II

0. 10 0, 18

+ "

0,8

0.16

^ II

0, 12 0, 14

+ "

0, 10 0. 18

V6/

- 1

II 1 II

+ "

II 1 II

0, 8 0. 16

__ II

41 1 14

0, 12 0. 14

+ "

0. 10 0, 18

_ II

0. 8 |0. 16

+ "

0, 12 0, 14

2V2/

+ 1

0. 13:0, 8

0. 14 0. 9

0, 150. 10

0, 16;0. 11

0, 17;0. 12

0. 18jO, 7

Appendix.

In the case of the diamond lattice the co5rdinates of position of the
surrounding atoms, with respect to a selected atom at the origin of the
coordinates, were given^ in Table I., page 264. In a similar manner the
following Table IV. gives them for a simple cubic lattice like that of rock
salt. The atoms are numbered as in the diagram Fig. 3.

The calculation of the numerical values of the functions /« and/e, given
in equations (i) and (2) above, has been carried out for each of the 124

» Loc. dt.

VOL.X.1

No. S. J

THEORY OP CRYSTAL STRUCTURE.

457

atoms surrounding the central atom in a cube of edge 4/, giving the results

in Table V. Each atom in the (jc, y) or s = o, plane gives F, = o

because n ia a factor of (i) for these atoms, and is zero in the plane

» = o.

Table V.

Odd Planes.

Plant.

Atoms.

-1

II
II
11

-3

II
-5

1, 2, 3.

4. 5. 6.

7-12.

13-18.

8, 9. 10.

2-7.

1. 2, 3.

F.-e« I +0.57735 S(a^)S(a^)/-* -5.052 S<»»S<»»/"*- • • }*•

p P' p p*

'• " +0.203703 " +0.27778
" *' +0.0413 " -0.0974
*' " -0.02165 ** +0.005846
" " -0.166667 " +0.277778
" *' -0.021395 " +0.00891
•• '* 0.000000 " -0.05385
" " -0.01136 '* -0.0126

(45)

(46)
(47)
(48)
(49)
(50)
(51)
(52)

Sum of odd planes ....

•* '* +0.6013 •* -4.6455

(53)

Even Planes,

-2

II
II

-4

-6

1, 2, 3.

4. 5, 6.
7-12.
13-15.
1, 2, 3.
4, 5. 6.
1.

F.-e« I -0.66375 S(<i/?)S(<i/?)/"*+0.8625 S<»»S«»/~*- • • l^.

p p* p p*

" *• +0.10821 " -0.03375

" •* -0.02838 " +0.0667

•• " +0.01273 " +0.00434

'* " +0.0960 " +0.0529

" '* -0.009569 " -0.016445

" " -0.0104125 *• +0.0043402 "

(54)

(55)
(56)
(57)
(58)
(59)
(60)

Sum of even planes. .

•• " -0.49517 " +0.94059

(61)

Sum of ode
planes. . .

i and even

" " +0.1061 *' -3.7049

(62)

The forces in the Table V. are given for the negative planes only, those
on one side of the central atom, thus including but one half of the cube
of edge 4/. In this form of lattice it happens that each atom is a center
of symmetry, not only for the position and kind of atom, but for the
direction of its axis of rotation as well. It always happens that there is
an atom of chlorine in rock salt, for example, having its axis parallel to
a corresponding atom of chlorine at an equal distance on the opposite
side of the selected atom. Equation (i) shows that the force that one
pair of such atoms exerts upon the selected atom is zero irrespective
of the value of /. The force changes sign if we reverse the signs of /, m
and n, the direction cosines, a remaining the same. The size of the space
lattice is, therefore, indeterminate if we consider the whole cube. This
makes it possible to impose a second condition before the size of the lattice
is fixed. This is, that we must obtain the same value of / whether we

45 S ALBERT C. CREHORE.

select a sodium or a chlorine atom upon which to figure the force. We
get the same value of / by equating to zero the total force of all the sodium
atoms in one half the cube, acting upon a selected chlorine atom, or by
equating to zero the total force of all the chlorine atoms in one half the
cube acting upon a selected sodium atom. Since the odd planes always
contain atoms of the opposite kind from the selected atom, it is concluded
that the equilibrium condition for all crystals like rock salt is obtained
by equating (53), which is the same as (6) to zero.

Degree of Approximation.

Had we included in the calculation the innermost cube only, of edge
2/, we should have had for the odd planes only the forces given in (45),
(46) and (49), and for the even planes only (54), giving the results

Odd planes, F, = ^{ + o.6i44S(a/3)S(a/3)/-^

-4.4964So»So*^'---}fc, (63)
Even planes, F, = «*{- 0.663755(0/3)2 (a0)f-*

+ o.8625Sa*2a*/-« •••}*, (64)

which may be compared directly with (53) and (61) above. The in-
clusion of the next outlying cube of atoms makes less difference here in
the case of the odd planes than in that of the even. But, the question
arises how much the results in (53) and (61), which have been used in
the paper, would be changed if we had included more of the outlying
atoms. This question can not be answered exactly until such calcula-
tions are made. They involve considerable labor and have not been
carried out beyond the cube of edge 4/, so that it is very desirable to obtain
some kind of an estimate of the error made in neglecting the more distant
atoms. To help give some idea of this error the forces due to all the
atoms lying along certain radii to infinity have been obtained in the case
of the even planes (61), which showed the greater difference when the
outer cube was included.

For example, along the radius vector from the origin to atom (— 6, i)
we come to atoms of the same kind and with axes parallel to the atom at
the origin at equal intervals, the second being twice, and third three times
as far, and so on, as atom (—6, i), the force of which is given by (60).
Since the first term varies as r"^, we may use this equation to find the
force of all other atoms along this radius to infinity. This is equivalent
to multiplying (60) by the sum of the series to infinity

Vot.X.1
Nas. J

THEORY OP CRYSTAL STRUCTURE.

459

»-4

+ 3-' + 4-* + 5

-4

0.082323,1

to obtain the coefficient of the fourth power term due to all other outlying
atoms along this radius. Similarly, for the sixth power term, we have to
sum the series

2-« + 3"* + 4"* + 5"* • • • = 0.017343.

This result would have made equation (60), if we include all atoms along
this radius to infinity, have the coefficient of /"* — 0.0008565, and of
^^ 0.0000755, different from that given in (60).

In a similar manner all atoms along several different radii have been
included, giving the following amounts to be added to the coefficients
as given in the Table V.

Dlr.ctioii of Raditt*.

Coef. of /-<.

Coef. of M.

(-6, 1)

-0.0008565
0.008903
0.0007875
0.00969
0.001425

0.0000755

(-2, 4) (-2, 5) (-2. 6)

(-4, 4) (-4, 5) (-4. 6)

(-2,1) (-2, 2) (-2, 3)

(-4. 1) (-4. 2) (-4. 3)

0.0005853
0.0002852
0.001240
0.00007606

Sum

0.001006

0.00052106

When these values are added to the coefficients in (61), it changes the
fourth power coefficient by about one part in five hundred, and the sixth
power by about one part in two thousand. The atoms added by this
process do not include all, even in the next adjacent cube with edge 61,
but these figures indicate that we can not rely upon the accuracy of the
figures in the Table V. in the third decimal place in the fourth power
coefficient, and it is not now possible to say whether the value will turn
out to be larger or smaller than those in (61), if we were to include all
surrounding atoms to a great distance. The preceding, however, may
be regarded as showing something as to the order of the error made by
omitting outlying atoms at greater distances.

The Diamond Lattice.

The diamond form of lattice consists of two interpenetrating face-
centered lattices. If we number the (iii) planes consecutively, from o
for the selected atom as before, the even planes in one of the lattices are
symmetrical with respect to the selected atom, giving a total force of
zero irrespective of the size of the lattice. The odd planes are not
symmetrical with respect to it, and the force must be calculated for all
odd planes, both positive and negative since these do not cancel each

> For a table of the sums of these series, see De Morgan's Calculus, p. 552.

460 ALBERT C. CREHQRB.

Other. In the case of ZnS the planes alternate, even planes being zmc
and odd sulphur, if we select a zinc atom on which to figure the force.
The equilibrium condition must be the same whether we select a zinc
or a sulphur atom upon which to figure the force as in the case of rock
salt.

The crystal CaFs has a structure similar to ZnS in some respects. If
Ca is substituted for Zn and one of the F's for S, the arrangement so far
is the same. The Ca planes recur at regular intervak, this interval
being the altitude of the tetrahedron on the elementary triangular base
in the Ca plane. One of the F planes occurs at one quarter of this distance
above the Ca plane, similar to the S in ZnS, and the other F at the same
distance below the Ca plane, which has no corresponding plane in ZnS.
The spacing of the planes then gives the following regular succession

F-Ca-F-Space-F-Ca-F-Space-F-Ca-F-Space and so on,

there being three planes at regular intervak and then one missing.
From this it is evident that if we select a Ca atom upon which to figure
the force, the positive planes will just balance the negative planes, thus
making the size of the lattice indeterminate. For the equilibriiun con-
dition, therefore, we do not need to figure the force upon a calcium atom.
The case is different if we select one of the F atoms, for, then the Ca atoms
are not symmetrical about it. On the other hand, the F atoms are sym-
metrical about it. This narrows the condition down to precisely the
same formula as applies to the diamond or zincblende, because we only
have to take into account the force of all the Ca atoms upon a fluorine
atom.

X^^] TOTAL EMISSION OP X-RAYS. 46 1

AN EXPERIMENTAL INVESTIGATION OF THE TOTAL
EMISSION OF X-RAYS FROM CERTAIN METALS.

By C. S. Brainin.

THE experiments described in this paper were undertaken with the
purpose of studying the integral intensity of the emission of x-rays
by metals from two standpoints: (i) The variation of the intensity of the
radiation from a given metal with the voltage and (2) the dependence of
the intensity upon the atomic weight (or number) of the metallic radiator
at diflferent voltages. It seemed that the use of the Coolidge cathode
and the gas-free x-ray tube gave opportunity for the maintenance of
unchanging conditions such as it was hitherto impossible to obtain with
the gas-filled tube. Moreover^ beside this important consideration, the
data upon which rest our ideas of the dependence of the total energy of
emitted rays upon the voltage and the atomic weight do not seem to be
founded upon experiment sufficiently extensive to preclude the desira-
bility of further investigation upon this important topic.

The experiments upon which has been based a relation between the
total intensity of x-rays emitted by a metal and the voltage across the
electrodes of the tube are mainly those of Whiddington and Beatty.
Their results are in agreement with the conclusion reached by Sir J. J.
Thomson.^ His theoretical investigation, founded upon certain atomic
assumptions and upon the Stokes ether pulse or stopped electron hy-
pothesis, brought him to the relation that the intensity of the x-rays
produced by the collisions of electrons and atoms should be proportional
to the fourth power of the velocity of the moving electrons. It should be
proportional, therefore, also to the square of the difference of the potential
impressed upon the electrodes of the tube. Furthermore, the results of
the above mentioned experimenters in conjunction with those previously
obtained by Kaye seemed to indicate that under like conditions of current
and voltage the emission of two metals is very nearly directly propor-
tional to their atomic weights. Both these results are summed up in the

equation

E = KAP^, (i)

where X is a constant, A the atomic weight of the radiating target and P

» Phil* Mag.. June, 1907.

462 C. S. BRAININ. [ggg?

the difference between the electrodes of the x-ray tube. Recently Bergen
Davis^ has deduced this formula, basing his development upon the
quantum hypothesis of radiation and assuming the x-rays to emanate
from the atom itself and not from the impacting electron. Equation (i)
is to be taken as applying only to the general or independent (non-char-
acteristic) x-radiation. It is then in place to examine the experimental
evidence so far brought forward in favor of this very broad formula.

Whiddington* makes the statement in an article on the ** Production
of Characteristic Roentgen Radiations " that " it comes out that E
(per unit cathode ray current) is nearly proportional to the fourth power
of the velocity." No experimental data connected directly with this
question could be found in any of his papers. At best he was getting
primary x-rays from one metal (silver) only; his range of voltages was
from 7 to 21 kilovolts. His studies on the other metals were made with
secondary rays produced by the primary rays from the silver anticathode.
Beatty* used a method which was quite direct; he measured the total
ionization produced in a chamber which absorbed completely the x-rays
that entered it at the voltages used by him. His range was from 8.5 to
21 kilovolts and he investigated four metals, rhodium, silver, copper and
aluminum. For the voltages used he found that rhodium, silver and
aluminum held well to the voltage square law; his results for copper show
a large increase at a voltage approximately corresponding to the critical
voltage for the characteristic X-radiation from copper but hardly justify
the acceptance of a straight line relation up to this critical voltage. In
the case of aluminum the entire voltage range is of course above the
critical voltage for its X-radiation. Rutherford^ took up this question
after the appearance of the Coolidge cathode and used a regular (com-
mercial) tungsten target x-ray tube. He compared the emissions at
three voltages, 48, 64, and 96 kilovolts and found that though the radi-
ated energy was very nearly proportional to the square of the voltages,
at the highest voltage the ratio was perceptibly greater.

As to the other factor contained in the equation (i), the atomic weight
A, the direct evidence in its favor is again mainly the above-mentioned
paper of Beatty and the work of Kaye.' Beatty found that the emissivity
of rhodium and silver as well as that of aluminum were in the correct
ratio in spite of the fact that the critical voltage for the X-radiation of the
latter had been exceeded. Kaye studied the emission of a great many

» Phys. Rev., Jan., 191 7.

* Roy. Soc. Proc., 85 A. 191 1.
•Proc. Roy. Soc., 89 A, 1913-14.
*Phil. Mag., Sept., 1915.

• Phil. Trans. Roy. Soc., 209 A, 1908-9.

^^^'] TOTAL EMISSION OF X-RAYS, 463

metals and found that upon partially screening out the soft (character*
istic?) rays the intensity of the remaining rays was approximately pro-
portional to the atomic weights. This work was done at a voltage of
about 25,000. No attempt was made to obtain complete absorption of
the rays in the ionization chamber. However, the experiments certainly
show the preponderance of the atomic weight as a factor in determining
the emissivity of the substances at a given voltage. It is also interesting
to note here that for voltages from 1,500 up to 3,500 volts Whiddington*
found that no connection existed at all between emission and atomic
weight.

A third point of interest which necessarily comes out in study of
the variation of the x-ray emission with voltage is the behavior of
the metals when the critical voltage for the appearance of the character-
istic rays is reached. With the exception of aluminum as noted above in
Beatty's paper, all the metals thus far studied, whose critical iT-voltages
lay within the experimental range, show a sharp increase in emissivity
when these rays are excited.

The Experiments.

A great difficulty attended the choice of substances for these experi-
ments, as it was badly limited by the necessity of having the element in
metallic form and of having the temperature of fusion high enough to
withstand the great heating to which they would be subjected.
Finally the following six were chosen, so as to obtain the greatest
range of atomic weights, and used throughout the work: Plati-
num, tungsten, silver, molybdenum, copper and cobalt. In order
that the intensity of ionization might truly represent the intensity of
emission a total absorption of the x-rays which, at any voltage, entered
the chamber, was desired and provision for it made as described below.
To avoid appreciable absorption of the x-rays in the wall of the bulb,
it was planned to have the rays pass only through a very thin window of
mica, before entering the ionization chamber so that the eflfect of this
absorption could be neglected even at low voltages.

Description of Apparatus, — Fig. i gives a diagrammatic view of the
very simple apparatus used. The Coolidge cathode, C, was mounted in
a horizontal and the anti-cathode, i4 , in a vertical position. The target
itself consisted of a brass block 2.5 cm. high and of hexagonal cross-section,
to each facet of which a different metal wac attached by means of tiny
screws at top and bottom. As the little rectangular metallic targets
were of different thicknesses, the brass block was trimmed down so that

> Roy. Soc. Proc., 191 1 A, 85. p. 99.

464

C. S. BRAIlflN.

[

all the surfaces were at the same distance from the center. Into the base
of the brass block was set one end of a narrow brass tube to which a bar
of soft iron was fastened, making a right angle with the tube. This tube
slipped over an aluminum rod, set up vertically in the x-ray bulb until
an iron pin, which was affixed to the upper end of the aluminum rod,
touched the inside of the brass block and thus provided a pivot for the
latter's rotation. The soft iron bar was cut in two places and hinged
there so that it could be folded up along the brass tube and the whole

>CM'

Fig. 1.

passed into the x-ray bulb through a wide glass tube fused in at the top.
When this rotating target had been properly slipped over the aluminum
rod, the wings of the soft iron bar were unfolded and the wide glass tube
entrance sealed off. The adjustment had previously been made so that
the centers of the surfaces of the target came opposite the cathode itself.
The anti-cathode could now be rotated from without the tube by means
of an electro-magnet. A mirror and a small sighting tube with cross-
hairs were used to insure the turning of the surfaces so that the x-rays
measured would come oflF each surface under like conditions. The
figure shows the position of the target with respect to the cathode and the
path into the ionization chamber.

The x-rays passed only through a mica window of thickness .001 cm.
before entering the ionization chamber. This thin sliver of mica was
fastened with De Khotinsky cement to a small lead plug which itself was
cemented to the large lead stopper. This in turn fitted into a glass tube
fused to the x-ray bulb, the end of which was slightly flared. The lead
and the glass were carefully fitted and cemented together. The whole
had been previously adjusted so that the channel for the x-rays through
the lead pointed toward the area from which the rays were to come.
The holes through the small lead plugs were .18 cm. in diameter and had
been so calculated that the radiation would traverse the length of the
ionization chamber without coming in contact with the side walls. The

Yl^^'] TOTAL EMISSION OP X-RA YS. 465

axis of the ionization chamber had to coincide with the axes of the holes
through the lead in order to effect this and the proper adjustments were
made before the apparatus was assembled.

The ionization chamber itself consisted of a thin-walled steel tube of
outside diameter 10 cm. and length 250 cm. This was used as a grounded
electrode; the other electrode (£, Fig. i) was a lead pipe stiffened by an
iron rod within it which ran the entire length of the chamber. It was
suspended inside the steel tube from four amber plugs, one of which carried
a wire connection out to the electrometer. It was finally decided to use
a Braun electrometer with an aluminum needle and measure the rate of
leak over a definite range of 500 volts. The needle was always charged
so as to indicate over 1,500 volts and the time was taken between 1,500
and 1,000 volts. As no readings were made with the difference of po-
tential between the electrodes less than 1,000 volts, saturation conditions
at all times were obtained in the tube. The electrometer needle showed
no tendency to stick and gave results which agreed well with one another.
The natural leak was practically always a negligible quantity. Proper
heavy lead shielding was placed around the x-ray bulb to prevent effects
in the electrometer due to secondary or reflected rays.

The x-ray bulb was evacuated with a mercury condensation pump of
the Langmuir type made entirely of glass after a design by Prof. G. B.
Pegram; it was used in conjunction with an oil fore- vacuum pump and
gave extremely rapid service. A freezing-out chamber was fused in
between the pump and the x-ray bulb and was kept immersed in a mixture
of salt and ice. This served to keep the mercury vapor pressure in the
bulb low enough to prevent the formation of a gas discharge. It took
several days of alternate pumping and running of the discharge to remove
the occluded gases and to make the bulb serviceable for the experiment;
but once it had reached a satisfactory state it was found necessary to do
but little pumping and several days of observations in succession were
sometimes possible without any need of using the pump. In spite of
the three De Khotinsky seals, made under difficult conditions, there was
at no time a real leak perceptible and most of the runs of observations
were made with the pump in readiness but not in actual use. Early in
the stage of preparation of the tube it was filled with hydrogen and when
this had been nearly all pumped out again a discharge was run for some
time with the anti-cathode acting as a cathode. This served to clear
perfectly the surfaces of the metal targets, some of which had become
slightly oxidized.

The current for heating the tungsten filament of the cathode came from
a storage battery placed upon a carefully insulated glass support. Four

466 C. S. BRAIN IN. [^SS

six-volt 2 J ampere batteries were used, two parallel sets, each composed
of two batteries in series, giving twelve volts. The electronic current
through the x-ray tube was measured by means of a Weston milliammeter
reading directly to tenths of a milliampere. One terminal of this meter
was connected to the anticathode and the other grounded. A strip of
tinfoil, which was grounded, was fastened around the x-ray tube near the
connection of the anticathode and the milliammeter so as to intercept
any possible leak over the surface of the glass and thus prevent it from
being registered on the ammeter. The main circuit high potential current
was obtained from a transformer fed by a 500-cycle, 150-volt alternator.
The current from this was rectified by a kenotron system. A very large
condenser was placed in parallel in the circuit so that the voltage could be
relied upon to be constant and the current unidirectional. The middle
of the transformer was grounded. The voltages were measured by means
of a spark gap between spheres 12.5 cm. in diameter.

Method, — ^The data taken in these experiments were obtained in two
distinct ways:

1. With a given metal acting as target the voltage was varied by small
steps and the rates of the electrometer leak measured; in this way the
ionization produced by the energy in the x-ray output of a particular
metal could be obtained as a function of the voltage.

2. Keeping the conditions of the voltage and current constant, the
different metal targets could one after another be subjected to the bom-
bardment of the electrons and the relative intensities of the emission of
x-rays for any particular voltage could be obtained. This was done for a
great number of different voltages throughout the range found possible
with the apparatus.

The lower limit of usable voltages is naturally determined by the
amount of energy necessary to produce readable leaks on the electrom-
eter in a reasonable length of time. It was found impossible to use
such high currents through the tube as are possible with the tungsten
Coolidge tubes prepared by the General Electric Company. A current
over 5 milliamperes soon broke down the vacuum even at very low
voltages. The upper limit in an experiment of this nature depends upon
how far the voltage can be raised and practically total absorption still
be obtained in the ionization chamber. This is really best determined
from the results themselves and the shape of the curves obtained by the
above method (i) is the best guide as to how high a voltage can be used.

In order to increase the absorption of x-ray energy in the chamber, it
was planned to use some dense vapor and the experiments were begun
with the chamber containing air saturated with ethyl bromide vapor.

sS:'^'] TOTAL EMISSION OF X-RAYS. 467

However, the study of certain phenomena which were found and are
hereinafter described made it desirable to have the air alone present, in
order to avoid the great increase in ionization which takes place in the
vapor when the x-rays absorbed by it contain wave-lengths which cor-
respond to the characteristic jRT-radiation from one of the component
elements of the gas. These would begin to appear, of course, when the
voltage across the tube is equal to the critical voltage for the excitation
of this radiation. This increased ionization. is entirely out of proportion
to the increased energy of the x-radiation due to the increased voltage
alone, and difficult to correct for. With the long ionization chamber
containing only dry air it was found that the highest voltage at which
the results could be relied upon was in the neighborhood of 33,000 volts.
Above 35,000 volts there appeared on all the curves a sudden great
increase due to the impinging of a large amount of the ionization upon
the end wall of the chamber. Tests with an electroscope showed that
no appreciable amount of x-ray energy remained unabsorbed below that
voltage. The lowest it was found possible to use was 4,700 volts, though
in general the readings were taken above 7,000 volts.

During the progress of the experiments the first x-ray tube set up
became so blackened that its further use was very difficult and at approxi-
mately the same time the glass pump cracked at one ef its fused joints.
The whole apparatus was, therefore, taken down; the pump was repaired
and a new tube set up differing only in slight details from the former one.
Data were then obtained on the metals which had not been examined
and much of the data already obtained was carefully gone over. It
was found that while the actual quantitative data differed by a constant
factor, the general results remained exactly the same.

Experimental Results.

In the results presented belo^ in the form of curves, the energy, as
represented by the rate of leak of the electrometer reduced to unit electron
current, through the x-ray tube, is plotted as ordinate and the square of
the voltage used is plotted as abscissa. If the resulting curve is a straight
line pointing to the origin, the energy of emission is really proportional
to the square of the voltage as expressed by the equation (i). A devi-
ation from the straight line passing through the origin and before the
critical voltage for the characteristic radiation is reached necessarily
means a failure of the equation to represent truly the phenomenon.

Platinum and tungsten^ the two metals of highest atomic weight in the
list and the only two whose critical voltage for the L-radiations are within

468 C. S. BSAINJN. ^°gg

the range of the experiments, show a similarity in that they both deviate
from the " square of the voltage " law. It is to be noticed particularly
that these curves do not become straight lines through the origin until
the voltage of 22,000 volts, approximately, has been reached in the case
of platinum, and 19,000 volts, approximately, in the case of tungsten.
It is to be noticed also that for a short
interval of voltage the radiation of tung-
sten seems to be more powerful than
that of platinum, a fact which is brought
out again later in connection with the re-
sults from method (2).

Molybdenum, as shown on Fig. 2,
seems to hold well to the law for the
range of voltages given and, not only
that, but also appears not to deviate from
this law upon the voltage reaching and
passing the critical voltage for its char-
acteristic X-radiations. For molybde-
num this is about 20,000 volts. All the
curves on Fig. 2 are mean curves based
Fv- 2. . on a number of separate sets of obser*

vattons on each metal.
Copper and cobalt also show a striking similarity in behavior and they
are plotted together in Fig, 3. Especially in the case of copper is it
possible to say that a straight line really represents the data up to the
point where the curvature, due to the characteristic, begins to be notice-
able. For cobalt also the curve may be said to be fairly close to a straight
line. The critical voltage for cobalt is about 10,000 volts ; that for copper
is 1 1 ,000 volts very nearly. Upon approaching these volt^es both curves
bend gradually upward and show what we may be allowed, for the sake
of brevity, to call a striking Increase of " efficiency." This increase is
greater for the metal of lower atomic weight, cobalt, at least within the
range of the experiment.

The results for silver are shown separately in Fig. 4. The upper curve
was obtained with the air in the ionization chamber saturated with ethyl
bromide vapor, the other with air alone, without this vapor. Upon
plotting the very first results obtained with silver, when the chamber
still contained the vapor, and calculating the ratios between the ionization
intensities and the squares of the voltages, the writer was surprised to
find the slight downward break in the curve as shown in the figure, upon
reaching the vicinity of the critical volt^e for the silver J^-characteristic

X^j*'] TOTAL EMISSION OP X-RAYS. 469

radiations, 25,000 volts. Up to this point the curve is surely a straight
line pointing toward the origin. To make certain of the real existence of
this phenomenon, the data for silver were taken several times alternately
with data for metals already examined, copper, platinum and molyb-
denum. In each case the results were a repetition of the data previously
obtained. Then the ethyl bromide vapor was removed from the ioniza-
tion chamber and dry air only used as absorber. The curve of the results

Fig. 3. Fig. 4.

so obtained shows the break much more prominently. The lessening of
the effect in ethyl bromide vapor was no doubt due to the reaching of a
voltage greater than the critical voltage for the X-characteristic radiations
from the bromine atoms in the vapor. This voltage is about 17,000. In
this case the increase in relative ionization power served to mask partly
the decrease in the so-called efficiency of the silver radiating mechanism.
Nothing could be found in the apparatus to explain away the anomaly,
which would not apply equally well to the other metals by which, how-
ever, it was not shown, Beatty and Whiddington had not found this
phenomenon because their highest voltages were below the voltage at
which it appeared.

II.
The results obtained from the second experimental procedure are given
in Fig. 5 below. The actual voltages and not the voltage squared are
here used as abscissa ; the ordinates represent the ratio that the emission
energy of the particular metal bears to the emission of molybdenum at
the given voltage. Molybdenum was chosen as the unit of reference
because it gave (in Fig. 2 above) the straight-line relation between voltage
squared and emission intensity postulated by the formula E=KAP*
and was therefore best suited to show deviations from this relation. Its
own emissivity is naturally then represented by a straight line of unit

470 C- 5- BRA IN IN. ^2

ordinate. For a better understanding of the data presented by the curves,
a table is added giving the atomic numbers and weights and their ratios
to those of molybdenum.

Table I.

MiUI.

At-Wt.

»tlootAt.Wi.

At. No.

RBU<>ofAt.NB.

195

2.17

1.86

1&4

1.92

1.72

107

1.10

1.12

1. 00

1.00

.66

.69

.62

.64

Examination of'Fig. 5 brings out some interesting information and
further confirmation of some of the results discussed above. The devi-
ation of platinum and tungsten from proportionality in their emission
intensity to atomic weight (or number) is very striking. If they really
obeyed this relation exactly, the curves representing each would be a
straight line parallel to the volt-
age axis and to the line repre-
senting molybdenum and at the
proper distances from these.
However, at the lowest voltage
shown, 4,700, the relative emis-
sion is very much below this
and it rises rapidly with the
voltage, becoming actually the
required straight lines in the
Fig. 5. neighborhood of 20,000 volts.

At this stage the ratio of the
emission of platinum to that of molybdenum, from the curve, is
about 2.06 and that of tungsten about 1.93, which is closely in pro-
portion to the atomic weights. Again there is a short interval of voltages
where tungsten is a better radiator than platinum. We might also,
perhaps, call attention to the fact that tungsten seems to reach the
straight line relation before platinum and that the voltages at which this
is reached are very roughly twice the critical voltages for their respective
characteristic L-radiations.

Silver appears to give an amount of radiation which is very slightly
out of proportion to its atomic weight at the lower voltages, but falls
pretty well into line above 10,000; at the lowest voltages its emissivity
is as high as that of platinxun and higher than that of tungsten. This

^^'] TOTAL EMISSION OF X-RA YS. 47 1

was to be expected if silver maintains the straight-line relation between
emission and voltage squared, but the slight apparent rise with respect
to molybdenum itself cannot, of course, be thus accounted for. The
decrease of the ratio for silver above 22,000 volts is a further confirmation
of the curious downward break in the curve for this metal as shown in
Fig. 4. Between these two changes the ratio to the emission of molyb-
denum is 1. 13 nearly.

In the case of copper and cobalt the results are fairly well given by
straight lines parallel to the molybdenum unity line until the appearance
of the characteristic jRT-radiation energy. They both then increase
rapidly, crossing the curves for molybdenum and silver, finally appearing
to approach a limiting value, and do not increase indefinitely in relative
emissivity. In the region below the critical voltages for their jRT-rays,
the average of the ratios of their emissions to molybdenum are .71 for
copper and .64 for cobalt, which is in good agreement with the values in
the table. The constant ratio of 1.83, to which cobalt rises after 20 kv.,
has no easily interpretable meaning.

III.

Three distinct modes of behavior on the part of the metals, when the
voltage is reached which corresponds to their critical voltages for the
JT-rays, have so far been discovered :

1. Most elements increase rapidly in the *' efficiency " of their radiated
energy output which is evidenced by a break upward, when the plotting
of the curves is made, as in this paper; such substances are copper, iron,
selenium, nickel, zinc, etc., which have already been studied, and cobalt
which this paper adds to the list.

2. Aluminum was found early to be an exception in that it seemed to
show no such break at all but gives evidence that it continues on in an
unbroken straight line; as mentioned at the beginning of this article,
Beatty bases part of his confirmation of the law of proportionality of
emission to atomic weight upon the fact that aluminum maintains this
relation throughout the range of his experiments. Molybdenum may
also now be added to aluminum as an example of such behavior.

3. The behavior of silver, which decreases in " efficiency," with the
appearance of its characteristic JT-radiations.

It seems to the writer that the different modes of behavior recited above
are rather antagonistic to the theory that independent and characteristic
rays come from different sources of radiation. It has been sometimes
held that the former are sent out from the impacting electron and the
latter from the electrons of the impacted atom, in a similar way probably

472 C. 5. BRAININ. [i

as the line radiation of the visible spectrum. One may then attempt to
explain away the aluminum and molybdenum behavior by saying that
the characteristic radiations in the case of these metals do not contain
relatively as much energy as, for example, the characteristics of copper;
and it is possible that a very slight increase of energy may go unnoticed
in such experiments as these. On the other hand, the characteristic
radiation from both aluminum and molybdenum h^ve been found to be
not at all weak. However, the one example of silver, unless refuted by
later work, makes the above theory unlikely; for it would be rather
difficult to understand why the energy emission from the impacting elec-
tron should not continue in proportion to the fourth power of the velocity.
It seems to make it probable that the entire radiation comes from the
atomic radiating structure itself, and, furthermore, that the reaction of
this mechanism is not parallel in all atoms when their respective critical
voltages are reached. The behavior of the independent radiations in
the neighborhood of the characteristic lines of rhodium, as found by D. L.
Webster^ may be perhaps taken as a support in this direction.

Summary.

1. A study has been made of the total intensity of the x-ray emission
from the six metals, platinum, tungsten, silver, molybdenum, copper and
cobalt over a range of voltages extending from about 5,000 volts to about
33,000 volts. The relations between the energy radiated and the voltage
and between the energy and the atomic weight were investigated; this
includes a test of the validity of the equation E = KAP^.

2. Below 20,000 volts approximately the energy radiated by platinum
and tungsten was found not to be proportional to A and P*; above this
voltage, however, it was in agreement with the above equation.

3. Molybdenum obeyed this relation throughout the range of voltages,
and showed no deviation from this relation when the voltage was increased
above the critical voltage for the JC-radiation.

4. Copper and cobalt seemed to obey this relation below their critical
voltages for the jRT-radiations, but above those voltages their emission
increased more rapidly than is required by the " voltage squared law."

5. Silver also obeyed this law below the critical voltage for its char-
acteristic jRT-radiation, but above that voltage the emission energy in-
creased less rapidly than is required by the " voltage squsu-ed law."

In conclusion I desire to express my sincerest thanks to Prof. Bergen
Davis who suggested the problem and whose kindly interest was a con-
tinual source of help and encouragement.

Phoenix Laboratory,

Columbia University.

> Phys. Rev.. June, 1916.

VOL.X.1
No. 5- J

THE DIFFUSION OF ACTINIUM RADIATION.

473

THE DIFFUSION OF ACTINIUM EMANATION AND THE

RANGE OF RECOIL FROM IT.

By L. W. McKebhan.

Introduction.

THE diffusion of actinium emanation, and the distribution of the
active deposit to which it gives rise, have been the subjects of
numerous researches.* In no case, however, has the experimental ar-
rangement been simple enough in its geometry to permit easy calculation
of the distribution to be expected on the basis of the known phenomena
of gaseous diffusion and of radioactive recoil. In the work here reported
this simplicity has been of prime consideration. Values of the diffusion
coefficient and of the range of recoil in air have been obtained.

Apparatus.

The essential features of the apparatus are shown in Fig! i, which
represents two sectional views through the vertical axis of the diffusion
space. Two similar brass plates. A, A, are held at either of two fixed

A A

i^^^^^^H

O O O O O O Orf) o o o o

OOOGOOOCHOOOGO

FiR. 1.

> W. T. Kennedy, Phil. Mag. (6), i8, 744, Nov., 1909. J. C. McLennan, Phil. Mag. (6),
34, 370, Sept., 1913. H. P. Walmsley, Phil. Mag. (6), 36, 381, Sept., 1913. A. N. Lucian,
Phil. Mag. (6), 38, 761, Dec., 1914.

rSftCOKD

474 L, W. McKEEHAN, IS»

distances apart by the three strips of ebonite, £, £, and B, and main-
tained at a suitable diflference of potential (usually 200 volts) by a battery
of lead accumulators. The source of emanation, 5, is a uniform layer
of an actinium preparation spread in the trough, T (which forms a false
bottom of the diffusion space, adjustable in height), or in two similar
troughs fixed in the side boxes, C, C. When the latter are in use the
trough T is, of course, removed entirely, and the emanation enters the
diffusion space through the holes, H. The square collecting plates, P,
P\ are four in number, two filling the windows, W, in each plate, A.
Their inner surfaces are flush with that of A , and they are flanged on the
outside to prevent leakage of the diffusing emanation at their edges.

The pressure in the diffusion space is adjusted by placing the whole
apparatus in a bell-jar of large volume containing some PjOi and con-
nected to a suitable pump and MacLeod gauges. Conditions of pressure
and temperature are maintained constant for a time in excess of five
hours, so that radioactive equilibrium will be practically attained between
the emanation and its products. The pressure is then changed quickly
to atmospheric and the bell-jar removed, these operations requiring about
thirty seconds. The four plates, P, P', are then all removed at once,
and their activities measured in an a-ray electroscope of such dimensions
that the full ranges of the a-rays of AcCu and AcC\ are utilized. From
these activities the activities in the preceding steady state are computed,
using Xb = 3.18 X 10"^ sec~* for the transformation constant of AcB,
This value of \b was checked many times during the experiments.

Table I.

Essential Dimensions of Apparatus^ in CentimeUrs,

Collecting plates. hXw 4.00 X 4.00

Width of diffusion space 1 1.0

Thickness of Diffusion space, s 0.50 or 1.00

Depth of surface of source below lower edge of P 0.64, 4.64. or in side boxes.

Height of A above upper edge of P* 4.0

Theory.

The assumptions made as the basis of the following theory fall into
three classes. The first comprises those that depend upon the construc-
tion and use of the apparatus, and that can therefore be realized to any
desired degree of approximation. These are as follows: a steady state
of diffusion and decay is obtained; the plates are unlimited; the diffusion
is linear in the region to be considered; the electric field between the
plates is sufficient to insure saturation with respect to the charged atoms
of AcA\ the activity computed from the measurements is proportional
to the total quantity of AcB on the plate in the steady state.

SoTs^'] ^^^ DIFFUSION OF ACTINIUM RADIATION. 475

The second class of assumptions comprises the applications of well-
established physical laws to the phenomena discussed and these are
acceptable at least to the accuracy attained in the experiments. They
are as follows: the diffusion coefficient is independent of the concentration
of emanation atoms and the degree of ionization of the air, but is inversely
proportional to the air pressure; the recoil atoms formed from the ema-
nation all have the same range, which is inversely proportional to the
air pressure; the direction of recoil is random; the effect of the applied
electric field upon the path of recoil is negligible; atoms of A cA, regardless
of their charge, are adsorbed by any solid surface which they may reach
by recoil or by diffusion.

The third class of assumptions is composed of the hypotheses suggested
by the present study or aribtrarily introduced for the sake of simplifying
the theory. Each of these will be criticized in the discussion of the
experimental results. They are:

1. Only emanation is supplied by the source.

2. The path of recoil is a straight line.

3. The variation of the concentration of emanation atoms within a
distance equal to the range of recoil is negligible.

4. The recoil atoms formed from the emanation are all positively
charged at the end of recoil.

5. The decay of Ac A and the growth of AcB takes place without chang-
ing the relative number of atoms on opposing areas of the two plates.

Referring again to Fig. i , take the origin of co5rdinates at the middle
of the lower edge of the lower positive collecting plate, measure x upward
on the positive plate, and y perpendicular to it. The distance between
the plates being j, the planes y = o and y = 5 are the positive and nega-
tive plates respectively.

The volume concentration of emanation atoms is a function of x only
and in the steady state is

where X is the transformation constant of the emanation, Di its coefficient
of diffusion in air at unit pressure, and p the pressure. Then a number of
atoms \ps will, on the average, decay per second per unit volume at
(jc, y), and the recoil atoms formed may strike either plate or be stopped
in the gas, depending upon the values of y and of the range Ri/p (where
i?i is range at unit pressure), and upon the direction of recoil.

If a sphere of radius Ri/p is drawn with its center at (x, y), the number
of recoil atoms received per second on either plate by direct recoil from
unit volume at {x, y) will be equal to

47^ L. W. McKEEHAN. [

area of segment of sphere cut off by plate considered

total area of sphere

^ height of segment
'diameter of sphere*
Thus the positive plate will receive by direct recoil

and

•r ^ ^^
o n y > T*

The negative plate will receive all the rest, either by direct recoil, or by
the aid of the electric field, after a recoil ending between the plates.

Considering all values of x and y and the conditions of radioactive
equilibrium, the surface concentration of recoil atoms in the steady state
at the point (.r, o) on the positive plate will be

in which only positive values of the integrand can be taken, and at the
point (x, s) on the negative plate will be

"'-^A^'L^bT^A

with the same restriction upon the integration.

The total activity on the two lower plates of height h and width w will
be

A = kws I p«(/x,
where jfc is a proportionality factor, and that on the two upper plates,

»2A

therefore

A' = kws J pjx = Ae-"^^^^^^ ;

and

Iog,(^,) = Aj^.

The fraction of the total activity on the positive plates will be

VOL.X.

Nas.

THE DIFFUSION OF ACTINIUM RADIATION.

477

"■», 0 "T <'■«, » S Jo

Performing the integration we get

Ri-py

2Ri

dy.

^ " 2 4R1

and

/ =

4ps

., Ri

If /> = y.

.4. -*?1

if p >— .

Experimental values ol A/A' and of/ for a series of values of p and 5
should afford a basis for calculating Di and 2?i and the constancy of these
quantities would be a partial confirmation of the theory.

Experiments and Discussion.

Preliminary experiments showed that the distribution on the plates
was practically independent of the potential difference for pressures up
to atmospheric and for potential differences in excess of 120 volts, so

0 S'le

^
^

\
\

•

--.

. ^^^ ^^^

•

— •♦-<

-C-J

#C-^

300

Fig. 2.

irtn%

900
90

200 volts was used throughout. Calculation shows that at pressures
below 55 mm. of mercury, 90 per cent, of the positively charged atoms of
AcA left nearest the positive plate will, for this value of the field, have time
to reach the negative plate before they change to AcB, so most of the
experiments were performed at much lower pressures than this, the lower
limit reached being 0.3 mm. The probable error in a single experiment
is least at a pressure of about 10 mm. of mercury, but is within a few
per cent, throughout the range of pressure used.

478

L. W. MeKEEBAN.

To base calculations on the value off, it was necessary to show that in
the absence of an electric field the opposing plates receive the same amount
of active deposit. This was rendered difficult by the fact that the electric
fields due to small differences of contact potential between portions of
the same plate, or of opposite plates, makes the distribution quite irregular
when the intense applied field is removed. This irregularity was small
immediately after careful cleaning of brass collecting plates but increased
with time unless the surface was protected from chemical action (prob-
ably oxidation). Lacquered brass plates, and pure gold plates cleaned
by a sand blast proved permanently satisfactory in this respect.

ZjO

,oer

rO^

«r

•30

Fig. 3.

The results obtained for / as a function of p, and for log {A I A') as a
function of ^p, are shown in Figs. 2 and 3 respectively, a point on each
figure being obtained in each experiment.

Table II.

Summary of Experiments Used in Figs. 2 and 3.

Number of experiments 49

with s « 1.0 36

" 5-0.5 13

" brauw plates 27

" " " " lacquered brass plates 14

*• gold plates 8

*• '* " ** source 0.64 cm. below plates 25

•• 4.64 cm. " '• 8

•• •* " •* '* in side chambers 16

No's^l ^^^ DIFFUSION OF ACTINIUM RADIATION, 479

In Fig. 2 the best theoretical curves based on the same value of i?i
are shown as broken lines, the value of Ri chosen to fit the experiments
being 7.0 cm., from which the value of the range in air at normal
pressure and the mean temperature of the experiments, 2o°.7, would be
.0092 cm.

In Fig. 3 the straight line through the origin represents the theoretical
relation for -Di = 83, or for the coefficient of diffusion into air at normal
pressure Pi/760 = .109.

A study of the curve for log {A I A') shows that there is a disturbing
factor operating at the lowest pressures, causing a relative increase of
activity on the lower plates. This is attributed to the diffusion from the
source of other members of the actinium series besides the emanation,
that is, to a failure of assumption (i). Such an effect has been noted by
other experimenters^ and it was here found to be much more noticeable
when no field was applied, showing that some at least of this stray
material left the source while charged, perhaps by recoil from radio-
actinium. To prove that the field applied in the neighorhood of the
source had the principle effect on purifying the diffusing stream, one of
the plates A was divided into two parts by a horizontal ebonite strip, the
upper part containing the collecting plates. An electric field in the
lower part of the diffusion space was then found to make log {A I A') the
same as if the entire diffusion had taken place in the electric field. The
nature of the disturbance produced by this stray material shows that
its quantity diminishes more rapidly with distance from the source than
the possible values of X and Di would allow for an uncharged gas.* The
easy adsorption of the atoms of a normally solid product, especially if
carrying electric charges, is believed a sufficient explanation of this
apparent discrepancy, but the behavior of this material will be studied
farther in a different apparatus.

The curves for / agree quite well with the theoretical curves, but the
nature of the discrepancies indicates that the calculated range at unit
pressure would be higher at lower pressures, and for closer approach of
the collecting plates at the same pressure. A more serious difficulty
is met on examining assumption (3) in connection with the values of
2?i and Di derived above, since it is found that this assumption fails at
pressures considerably above the lower limit actually reached, and that
the curve for / should be much lower at low pressures than the simpler
theory indicates. A few computations have been made taking account

> L. Wertenstein. Ann. de Phys. (9), i, 347, April; 393, May, 1914 (in case of radium emana*
tion and products, also refers to previous work).
* Excluding Ac A, for which X is far too great.

Sboomo

480 X. W. McKEEHAN. [

of the actual variation in the concentration of emanation with the distance
from the source. The calculations are tedious, since

I du I del {sin<^cos««^-^^'^^»>-*"**«^)A^

_ 0 «/0 «/oo«-i(«/«)

r/»2ir /»coe-l(-jfp/-Si)

dy I de I {sin^-^^»'^>'*"*~'^}A^

•/O «/oot-i [(•—v)plBi]

+ 2 I du \ del {sin<^cos««^-*'^'^^»*"*°**«MA^

(only real limits for integration being allowable).

Although one of the integrations in one of the integrals can be per-
formed the resulting form is so badly adapted to valuation by quad-
rature that all six integrations had to be effected by that means.

Three such calculated values of / are shown by crosses in Fig. 2 and
refer to the same conditions as the lower broken line.

The agreement between the experiments and the approximate theory
now becomes a matter for explanation, and it is suggested that a moderate
tortuosity in the paths of the recoil atoms (i. e., abandonment of assump-
tion (2)) would revalidate the approximate theory by making it im-
probable that many atoms could reach their full range at low pressures
even if projected parallel to the plates. This would also raise the ap-
parent range at low pressures by making a greater proportion of all recoil
atoms reach the plates than would be expected on the hypothesis of
straight line recoil. Experiments by C. T. R. Wilson's method are in
progress to detect if possible whether photographs of recoil trails will
show that such a tortuosity exists.

A partial failure of assumption (4) would help to explain the observed
results, and this at first sight seems plausible, since the rate of acquire-
ment of the positive charge by the recoil atom has been shown to depend
upon the gas pressure.^ Since, however, the whole number of collisions
during recoil probably depends very little upon the pressure, the final
condition should be independent of that variable, and the values of / at
high pressures indicate that few recoil atoms are then uncharged, even
when the chances of so-called initial recombination are the best. That
there is some increase in recombination at high pressures is indicated
clearly by the high values of / at pressures above 100 nun. of mercury

(Fig. 2).

In this connection the high value for the fraction of the recoil atoms
uncharged, which was obtained by Lucian* with much greater applied

» L. Wertenstein, C. R., 161, 696, Dec. 6. 1915.
* Loc. cit.

No'sM ^^^ DIFFUSION OF ACTINIUM RADIATION. 48 1

potentials, is to be noted. It is due, no doubt, to the fact that the full
range of all the a-particles emitted by the collected deposit was not used ;
partly on account of the dimensions of the measuring apparatus, and
partly on account of the concavity of one of the collecting surfaces.
Saturation with respect to the active deposit atoms was also very dif-
ficult to obtain in the cylindrical apparatus employed, as the author
himself observes. Under these unfavorable conditions his value for the
fraction of all recoil atoms which appeared to be uncharged at atmospheric
pressure, was 5.1 per cent., which would correspond to a value of/, in
the notation of the present paper, approximately equal to 0.025 whereas
the value of/ here found at this pressure was only 0.015. The disagree-
ment is slightly greater if account is taken of the fact that the ratio of
volume to surface in the diffusion space was greater in Lucian's experi-
ments than in these. If there had been no initial recombination, how-
ever, the value of / for s equal to one centimeter would have been only
0.002 at atmospheric pressure, so the two results agree better with each
other than either does with the hypothesis that such recombination is
negligible.

The remaining assumption open to criticism, (5), is that the efficiency
of recoil of AcB from the collecting plates is low. The effect of any
failure of this assumption would be to increase again the value of /,
especially at low pressures, and thus to assist in explaining the agreement
between the experiments and the first theory proposed above. The
efficiency of recoil from a sanded surface is certainly low.

The experiments of Kennedy^ on the diffusion of emanation between
circular charged plates at various pressures and separations show several
of the effects here studied, and even give approximately the same value
of jRi (6.2 cm.), when his data is treated in the same way. The results
on diffusion coefficient are not directly comparable. The value here
obtained for this constant is, however, about the same as that found by
earlier investigators.* No satisfactory explanation for the large value
obtained, in comparison with that to be expected from the high molecular
weight that the emanation must certainly possess, seems as yet available,
except the probability that the inert atom of emanation has a weaker
stray electric field than the complicated molecules of comparable weight
on which the extrapolation of Graham's Law has been based.

1 Loc. cit.

* Kaye and Laby. Tables. 103, 191 1.

482 L. W. McKEEHAN, [^g

Conclusion.

The laws of ordinary gaseous diffusion and of radioactive recoil suffice
to explain the principal effects observed in connection with the diffusion
of, and deposit from actiniiun emanation. The range of recoil from this
emanation in air at normal pressure and at 20.7** C. is about .0092 cm.
(dz 2 per cent.), the diffusion coefficient under the same conditions is
about .109 (dz 2 per cent). Further work on some disturbing causes is
proposed.

I take great pleasure in thanking my colleagues, especially Dr. J. T,

Tate, for suggestions and constructive criticism throughout the course

of this investigation.

Physical Laboratory,

University of Minnesota,
June 14. 191 7.

No's'^] ^^^ PRESSURE INCREASE IN THE CORONA. 483

THE PRESSURE INCREASE IN THE CORONA.

By Earlb H. Warnbr.

I. Introduction.

IT has been reported by Farwell and Kunz that at the instant the
corona appears about an axial wire in a cylindrical tube, the pressure
of the gas in the tube suddenly increases.^ It has always been stated
that this pressure increase could not be due to heat, because of the in-
stantaneous character of its appearance, and because of the rapidity
with which it disappears as soon as the potential is removed from the
wire. Since the only theories which have been advanced to explain
the corona assume it to be an ionization phenomenon, it seemed reason-
able to suppose that this pressure increase was due to the increase in the
number of gas particles in the tube, and so it was called ionization pres-
sure. Experiments have been performed and reported^ which show that
this pressure increase is exactly proportional to the corona current, with
the wire positive when dry air, hydrogen, nitrogen, carbon dioxide,
oxygen and ammonia are the gases in the tube. Since the publication
of this data Arnold' has contended that the pressure increase could be
completely accounted for as the result of Joule's heat, and that the
assumption that it is due to ionization is untenable. To support this
contention Arnold performed experiments *' by electrically heating the
central wire in apparatus similar to Farwell's and " observed the pressure
increase. With such an apparatus Arnold attempted to show (i) that
an increase in pressure due to heat appears suddenly, (2) that for a given
power consumed in the tube the increase in pressure due to heat is of
about ** the same magnitude as those observed " in the corona.

In order to show clearly that the pressure increase is not due to heat
a series of comparative experiments were performed with the pressure
increase caused, first, by producing the corona glow on the wire and,
second, by heating the central wire. The pressure increase observed in
the first sdt of experiments will be referred to as caused by corona and in
the second set as caused by heat,

» Dr. S. P. Farwell. "The Corona Produced by Continuous Potentials," Proc. A. I. E. E.
Nov., 1914. Dr. Jakob Kunz, "On the Initial Condition of the Corona Discharge," Phys.
Rbv., July, 1916.

*Earle H. Warner, "Determination of the Laws Relating Ionization Pressure to the
Current in the Corona of Constant Potentials," Phys. Rev., Sept., 1916.

» H. D. Arnold, (Abstract) Phys. Rev., Jan., 191 7.

484

EARLE H. WARNER,

Pr«

Oiw To m$m%.

.1.00

2 0.75
1 0.50
•"0.15

A few computations have also been made which strengthen the results
of the experiments.

II. Experimental Results.

1. The reason why one who sees this pressure increase, as recorded by
a quick-acting pressure meter, thinks it is not a heat effect, is because of
rapidity with which it appears and disappears. Arnold showed that the
pressure increase occurred quite rapidly when caused by heat. The
following curves show the difference in the rapidity of appearance and
disappearance of the pressure increase caused by heat, and caused by
corona. It will be noticed in Fig. i, where the pressure increase was

caused by heating the central wire, that
fifteen seconds was required for the
prssure to come to its maximum value,
and that from the time . the current was
broken twenty-five seconds was required
for the pressure to return to practically
its original value, while in Fig. 2, where
the pressure increase was caused by co-
rona, only three seconds was required
for the maximum pressure to be at-
tained and that the pressure came back
to practically its original value in eigh-
teen seconds. In this last case from the
appearance of the phenomenon it seems, if the aneroid pressure me-
ter had less inertia, that the pressure increase could be determined in
less than three seconds. These curves show that the
pressure increase appears five times as rapidly when
caused by corona as when caused by heat, and disap-
pears also more rapidly.

2. In the pressure increase due to corona, a short
time interval of five to seven seconds occurs after the
sudden increase of pressure, before the heat effect in
the corona begins to be noticed. This is shown by an
abrupt bend, i4, in the curve where the pressure in-
crease is plotted against time, as is done in Fig. 3.
No such bend occurs in the case where the pressure
increase is caused by heat alone, as is shown in Fig. i. In the work which
has previously been reported the pressure increase measurements were
always taken at the point A , and this seems to be practically independent
of the heat effect.

20 JO 40

Fig. 1.

50 (0 70

OlW To

Fig. 2.

VOL-Xl

Nas. J

THE PRESSURE INCREASE IN THE CORONA.

485

3. The heat which is produced in the corona discharge, shown by the
gradual pressure increase from B to C, Fig. 3, is distributed throughout the
whole volume of enclosed air and so, when the current is broken does not
radiate rapidly because the air is a poor conductor. This is shown very
clearly in Fig. 4. This seems to show that the pressure increase due to

Pr»M

Itmnmi^ Dim To e«reiw.

f? '9® J*?-.^^ '^ '«» 200
timm la ••o«ndc.

Fig. 3.

Fig. 4.

heat in the corona is represented by the difference of ordinates of C and
B (Fig. 4). As soon*as the corona current is broken at C the increase
in pressure due to corona at once disappears, but the increase in pressure
due to heat in the corona discharge remains, as is shown by the difference
of ordinates of D and A, This difference is always very nearly equal to
the difference of ordinates of C and B, This heat energy produced by
the corona current, since it is distributed through the gas, radiates very
slowly, as is shown by the gradual descent of the curve from D to E.
No such effect is observed when the increase of pressure is due entirely
to heat, as is shown in Fig. i. This curve (Fig. i) shows that twenty-five
seconds after the current through the wire is broken at C the resultant
pressure increase due to heat has practically disappeared; while Fig. 4
shows that twenty-five seconds after the corona is removed from the wire
the increase in pressure due to the corona has disappeared, but practically
all the pressure increase due to heat in the corona (ordinates C minus B
approximately equals ordinates D minus A) still remains and radiates
very slowly.

4. If the increase in pressure is due to heat, the same increase in
pressure should result when the same power is consumed (a) with a
corona current through the gas, (ft) with a heating current through the
wire. Figs. 5 and 6 show that this is not the case. The powers con-
sumed in the two cases are not exactly the same, but one can see that were
they the same, the increase in pressure due to corona would be approxi-

486

EARLE H, WARNER.

rSBCOMD

mately one half the increase in pressure due to heat. The power in the
case of the corona was obtained by multiplying the potential diflference
between the wire and the tube by the corona current, and in the case of

^ r

l)a» To H««t.

I'll

10 20 30 40

60 70 W

tim la

Fig. 5.

Zaeraaa* Do* T» Oorma.
0.2M Wfttta.

I I ' f ' 1 I

10 20 JO *0 50 §0 70

<0 90 100

tlw la-tM6nd*.

Fig. 6.

the heated wire was obtained by multiplying the current through the
wire by the potential difference across that portion of the wire which was
in the tube.

5. If the increase in pressure in the corona discharge is due to heat the
temperature of the air in the corona tube must increase. This may or
may not be the case in the luminous layer near the wire but the tem-
perature of the gas in the tube at a point four millimeters from the wire
actually decreases. This was determined by inserting a sensitive ther-
mocouple made of very fine Copper-Advance wire into the corona tube.
The temperature decreased only at the instant the corona appeared. In
a short time, after the heat due to the corona began to appear (corre-
sponding to the slope B to C, Figs. 3 and 4) the temperature of the gas
in the tube began to increase. This cooling effect is shown in Fig. 7.
Comparing Figs. 7 and 3 it is seen that the increase in pressure which

was measured at A was observed while there
I ^cooling Effeot In Corona, was an actual cooHng in the corona tube.
* Tia» In seo. This cooling should be expected when air or

oxygen are in the tube, for under these condi-
tions ozone is formed. Since the formation of
ozone from oxygen is always accompanied with
an absorption of heat the temperature of the
air or oxygen would tend to lower. Mr. J.
W. Davis, working on corona about hot wires
in hydrogen, has discovered that the appearance of the corona about a
tungsten wire heated to white heat, causes it to cool to dull red. This
tends to show that even in the corona glow itself there is a cooling effect.

6. If the increase in pressure in the corona is due to heat one should
expect it to be the same with the wire either positive or negative. As
has been previously mentioned it is impossible to obtain measurements

Fig. 7.

VOL.X.

No,

^^] THE PRESSURE INCREASE IN THE CORONA. 487

when the wire is negative because of the presence of beads. The negative
corona is entirely different from the positive corona.

7. The following consideration will further show that the increase in
pressure can not be due to heat. The heat produced by the corona
current will be given by the equation H = 0.238 eil and, if the observed
pressure increase is due to heat, the increase in pressure Ap will be pro-
portional to the heat, and we can write Ap = k eiL Now the only way
for Ap to vary directly as t, the corona current, as is the case — shown by
curves in the last article — is for « to be independent of i. Data shows that
this is not the case.

III. Results from Theoretical Considerations.

1. If the increase in pressure is due to heat it is possible to compute
the magnitude of the pressure increase when one knows the watts of
electrical energy consumed in the tube. The trial represented in Fig. 6
gives us this data. The observed pressure increase was measured in
three seconds so that the total number of joules of work consimied by
the tube in that time was 3 X 0.266 = 0.798 joules and this corresponds
to 0.1909 calories. Knowing the volume of the tube, the temperature
and pressure of the air in it, the mass of the air in the tube can be com-
puted. With the above-mentioned quantity of heat and mass of air,
together with the specific heat of the air at constant volume, the temper-
ature rise of the air can be computed, assuming that the electrical energy
is converted into heat. This temperature rise comes out to be 2.44° C,
which at constant volume corresponds to a pressure increase of about
nine cm. of water, while the observed pressure increase in this particular
trial amounts to about seven tenths cm. of water. In this computation
radiation and conduction losses have been neglected because they would
be very small from a body 2.44° C. above room temperature. This
shows that the observed results lie in a different order of magnitude from
what would be expected if Arnold's theory were true.

2. Arnold states, if "we compute the corona currents that would
result from the presence of enough ionized particles to produce the ob-
served pressure changes, the currents calculated are many thousand times
greater than those actually obtained." Such a statement is only true
when the ionized particles are produced in a uniform or practically uni-
form electric field. This is not the case in the corona tube. H. T. Booth
is publishing data on the distortion of the field in the corona tube. This
data shows that the potential gradient near the wire is very high — of the
order of 30,000 volts per cm. This is the arcing gradient, in which it is
probable every molecule is ionized. Then for a long space between the

488

EARLE B. WARNER.

wire and the tube there is a very small gradient. With this condition
of the field, near the wire every molecule may be ionized and still the
resultant current be very small, for few of the ionized particles near the
wire will pass through the space where there is a small gradient. Simple

computations based on kinetic
theory show that the maximum
observed pressure increases can
be explained by ionization if
every molecule of the air with-
in 1.39 nun. of the wire is
ionized. Within this distance
the potential gradient is equal
to the arcing gradient and
therefore probable that all mole-
cules are ionized.

llOD

xioo

IMO

1200

1000

IV. Further Verification
OF KuNz's Theory.

The final equation as pre-
sented in the last article is

Jfei = — (^1 — />o) + a constant,
e

where i is the corona current,
vo the volume of the tube, e
the potential difference between
the wire and the tube, pi — po
the pressure increase, k a con-
stant and po the initial pressure. This equation shows that for a con-
stant potential difference «, the current i should increase as po is low-
ered. Data were taken, by measuring the current at various measured
pressures, caused by a constant potential difference, which verifies this
theory. These data are shown graphically in Figs. 8 and 9 when pure
hydrogen and nitrogen respectively were the gases in the tube.

Fig. 8.

V. Summary and Conclusions.

Experimental results show:

1. That the increase in pressure due to corona appears and disappears
much more rapidly than when due simply to heat.

2. That the heat in the corona discharge is not a prominent factor
until many seconds after the corona appears.

Vol. X-l
No. 5. J

THE PRESSURE INCREASE IN THE CORONA.

489

3. That in equal energy experiments the increase in pressure due to
corona diflfers from the increase in pressure due to heat by about 50 per
cent.

4. That at the instant the corona appears the gas in the tube at a
small distance from the wire is cooled.

1400

laoo

1000

Oorrwrt As • JUMtlw of PrvMurs.
OMMtaat Tolt«c>

Vir* ♦

7392
Tolt*.

MO
9wmn la Mk

Fig. 9.

5. That the theory advanced by Kunz is verified in one more field,
namely in the relation between current and pressure for constant voltage.

These results together with conclusions drawn from simple calculations,
force one to believe that the pressure increase in the corona discharge is
not due to Joule's heat. With the recent knowledge of the distortion of
the field in the corona tube it seems very possible that the increase in
pressure is due to ionization.

The writer desires to express his appreciation to Professor A. P. Carman

for the use of the laboratory facilities, and to Dr. Jakob Kunz for his

continued interest and suggestions.

Laboratory of Physics.
University op Illinois,
June, 191 7.

490 A. LL. HUGHES. [toSS

THE EMISSION OF ELECTRONS IN THE SELECTIVE AND

NORMAL PHOTO-ELECTRIC EFFECTS.

By a. Ll. Hughes.

THE selective and normal photo-electric effects have been investigated
almost entirely through a study of the variation in the total number
of electrons emitted from suitable metallic surfaces, with the wave-length
of the light used. It is therefore desirable to attack the problem in
another way in the hope that some evidence as to the difference between
the two effects may be obtained. The author^ attempted this in an
investigation of the distribution of electrons emitted from a surface of
sodium-potassium alloy. The results showed that the distribution was
not identical for the selective and for the normal photo-electrons, but
did not settle whether the difference was one in the direction-distribution
or in the velocity-distribution of the photo-electrons in the two effects.
The following experiments show that there is a definite difference in the
direction-distribution of the photo-electrons.

The apparatus consisted of a glass tube about 5 cm. wide, provided with
three aluminum electrodes as shown. These could be connected to an
electrometer separately or together. Two small apertures, at opposite
sides of the cylindrical electrode C allowed a narrow beam of light to
pass in and out. The light was focused on to a small area (about 4 nmi.
square) at the center of the sodium-potassium alloy surface. A mercury
lamp was used as source of light. To secure light of the wave-length
corresponding to the maximum of the selective photo-electric effect, it
was passed through a Wratten blue filter (made by the Eastman Com-
pany) to isolate the blue lines of the mercury arc. The ratios of the solid
angles subtended at the center of the surface of the alloy, by the electrode
A, by A + B, and by A + B + C, were roughly as 15:55:100. The
tube was exhausted by the charcoal liquid air method and sealed off.

The method of experiment was to measure the number of electrons
received by A^by A -\- B, and by -4 + 5 + C respectively, when there
was no field to make the electrons deviate from their straight line paths.
It is not sufficient to connect the alloy and the electrodes to earth to
secure the absence of an electric field, the contact difference of potential
must be annulled. A very convenient way of doing this was suggested

» Hughes. Phil. Mag.. XXXI., p. 100. Feb.. 10.16.

VOL.X.

sf-]

EMISSION OP ELECTRONS.

491

by the work of MilUkan.* Let V be the negative potential which is
' necessary to apply to ^4 + B + C in order to stop the fastest electrons
emitted from the alloy when illuminated by (unpolar-
ized) light of frequency v. Then V + K is the total
potential difference between the electrodes A + B + C
and the alloy where K is the contact difference of poten-
tial. Let vo be the lowest frequency capable of causing
the emission of photo-electrons from the alloy. Then

we have

e(V + K) ^hv-hvo,

K -=' (v- vo) - V.
e

To get the lowest frequency capable of exciting the photo-electric effect, a
powerful carbon arc was used to illuminate the surface in conjunction
with several red and orange filters. A barely measurable effect was
obtained when the light was filtered through a thin molybdenite flake
which was opaque to light of wave-length shorter than X 7,100. This was
therefore taken as the long wave-length limit of the photo-electric effect
of the sodium-potassium alloy. A potential of .35 volt was sufficient
to stop the electrons due to the green line (X 5,461) of the mercury arc.
Applying this to the above formula, we get Jf = .31 volt. This is prob-
ably subject to an error of dz .05 volt, on account of some uncertainty
in the determination of the long wave-length limit. (It should be ob-
served that this value of the contact difference of potential between the
sodium-alloy and the aluminum electrodes is smaller than might have
been expected.) Thus to secure the absence of an electric field between
the alloy and the electrodes -4, B, and C, the alloy must be made .3 volt
negative with respect to them.

On illuminating the alloy with light polarized successively in the E\\
and in the E ± planes (that is, with the electric force parallel and per-
pendicular to the plane of incidence respectively), the ratio of the selective
to the normal effect foi the electrons caught by ^4, by A + B^ and by
A + B + C in turn were found to be as follows:

selective effect

Collecting Electrodes

Ratio

A +B +C,

A -hB

normal effect

10.3

11.7

17.5

Jt \BX)

These observations indicate a greater concentration of the selective

^ Millikan, Phys. Rbv., VII.. p. 18, Jan., 1916.

492

A. LL. HUGHES.

photo-electrons along directions near the perpendicular to the surface,
as compared with the normal photo-electrons. In the normal eflfect
the charges received by A, A + B, A + B + d were as 16:60:100
while in the selective effect they were as 27:81 :ioo. The normal photo-
electric distribution is therefore closer to that which would be obtained
on the supposition that equal numbers of electrons are emitted per unit
solid angle, regardless of direction (15:55:100). No attempt was made
to allow for reflection of electrons in this rough comparison. To make
sure that those results might not be due in some way to the field not being
zero, on account of an errorin estimating thecontact differenceof potential,
the observations were repeated twenty-four hours later with different
negative potentials applied to the alloy.

Collectiog Blectrode.

selective effect (£i\
normal effect \£±)'
Potential on the Alloy.

—.5 Volt.

—.3 Volt.

—.1 Volt.

A-\-B-\-C

9.6
11.1
13.3

10.2
11.5
13.8

10.1

A+B

10.7

14.8

These results show that slight departures from exact compensation of the
contact difference of potential do not affect the ratios to any great extent.
We may therefore conclude that the ratios really indicate a difference in
the direction distribution of the photo-electrons in the selective and
normal photo-electric effects. That the selective, photo-electrons tend
to crowd more along the perpendicular to the surface than the normal
photo-electrons might, at first sight, be expected, since the electric force
in the light has a component along the perpendicular to the surface.
Calculation shows however that it is impossible for an electron vibrating
about a position of equilibrium, to acquire energy of the order of that
possessed by a photo-electron, from the electric force in the light beam,
unless we suppose that the vibration is undamped and that the electron
can go on accumulating energy undisturbed, for over a million vibrations.
One then tarns to the view that there are vibrating systems, which, over
a certain range of frequency, are more easily broken up by alternating
electric forces (of the right frequency) perpendicular to the surface, than
by electric forces parallel to the surface.

The presence of a maximum on the curve connecting the number of
electrons emitted per unit energy of the incident light, with the wave-
length, has been taken to mean that the selective photo-electric effect is
a resonance phenomenon. As Pohl and Pringsheim^ have shown in the

» Pohl and Pringsheim, Verh. d. Deutsch. Phys. Ges., XV., p. iii. 1913.

fl^y'] EMISSION OF ELECTRONS, 493

case of one metal at least, the mere presence of a maximum may be
completely accounted for by a consideration of the depth to which the
light penetrates into the surface and the chances which the photo-elec-
trons produced at diflferent depths have of emerging. When the condi-
tions are arranged so that the light is absorbed in a very thin layer (that
is, by using a very oblique beam), so that all the photo-electrons released
in the surface have a greater chance of emerging, then the maximum
disappears. The real selective photo-electric effect as defined by Pohl
and Pringsheim, however, needs more than a maximum on the curve to
indicate its presence; the photo-electric effect associated with light
polarized in the E\ \ plane must be several times larger than that associated
with light polarized in the E ± plane, and also must possess a pronounced
maximum at some wave-length in the region where the maximiun appears.
Indeed, the maximum appears only in the photo-electric effect produced
by light polarized in the £|| plane. Moreover this maximum must
become more and more pronounced as the obliquity is increased, that is,
as the electric force in the light beam becomes more and more perpendic-
ular to the surface. Either the light polarized in the E\ \ plane is absorbed
in a very much smaller depth than the light polarized in the E ± plane,
with the result that the photo-electrons produced by light polarized in
the £|| plane escape from the surface in greater numbers; or else there
must be resonance systems in the surface which have the property of
responding only when the electric force in the light beam has a component
perpendicular to the surface. There is no evidence from optics to support
the first hypothesis. So far as the maximum emission velocity is con-
cerned, the work of Richardson and Compton^ and of Millikan* shows
that there is nothing unusual in the behavior of the photo-electrons from
sodium, even in the region where the maximum selective effect is ob-
served, when compared with other metals which give only the normal
effect. Hence on the second hypothesis, it would be necessary to
suppose that the special systems which give rise to the selective effect
are fundamentally of the same nature as those which give rise to the
normal effect. The selective effect would then be due to the fact that
there is an exceptionally large number of systems of a certain period so
oriented as to respond to light polarized so that there is an electric force
perpendicular to the surface.

The results obtained in this paper suggest a systematic examination of
the velocity distribution and the direction distribution of photo-electrons

> Richardson & Compton, Phil. Mag.. XXIV.. p. 575. 1912.

>Millikan. Phys. Rbv.. VII.. p. 355, March. 1916. Millikan and Souder. Proc. Nat.
Acad, of Sci.. II.. p. 19. Jan.. 1916.

494 ^- ". HUGHES. [sss:

emitted from surfaces illuminated by polarized light. It is proposed to
carry out the experiments on metals which show the selective effect such
as sodium-potassium alloy and on metals such as mercury which show
only the normal effect. By using liquid surfaces, we can be much more
certain that the plane of polarization of the light has a definite meaning
with respect to the plane of the surface.

These experiments were carried out in the Palmer Physical Laboratory
at Princeton during the summer of 1916. I wish to express my best
thanks to Professor Magie for placing the facilities of the laboratory at
my disposal.

Thb Rick Institute,
Houston. Tbx.

No's^l ^^^ IONIZING POTENTIALS OF GASES. 495

THE IONIZING POTENTIALS OF GASES.

By a. Ll. Hughbs and A. A. Ddcon.

INVESTIGATIONS on the least energy required to ionize molecules
of a gas by the impact of electrons lead to results which may often be
used to test theories of atomic structure. Such experiments are better
known as experiments on the ionizing potentials of gases. Of the recent
experiments on this subject, the best known are those of Franck and
Hertz.^ During the past year, the results of some of these experiments
have been verified and extended by Goucher* and Bazzoni.' Yet up to
the present, the ionizing potentials of only six or seven gases are known.
It was thought that a systematic investigation of the ionizing potentials
of a number of gases — compounds as well as elements — ^would prove
valuable. It was also thought worth while to measure the ionizing po-
tentials of some gases which have already been investigated to see
whether the modifications in the experimental methods lead to appre-
ciable changes in the published constants.

First Method.

In the method used by Franck and Hertz and by Goucher, the electrons
are accelerated by an electric field up to a gauze, and a certain proportion
of them pass through the interstices into another region where they are
subjected to a retarding field. The positive ions produced in this region
are driven into an electrode connected with an electrometer. One obvious
defect of this method is that, at potentials just above the ionizing poten-
tials, the part of the gas in which ions can be produced is limited to a thin
layer close to the gauze, for the electrons are quickly retarded to a speed
below which they do not ionize. A method in which all the gas can be
ionized, even when the applied potentials are close to the ionizing poten-
tials offers advantages in precision in fixing the ionizing potentials. The
apparatus shown in Fig. i approximately satisfies this condition. The
photo-electrons from a platinum disc P, illuminated by ultra-violet light,
were accelerated by a suitable electric field towards the hollow platinum
cylinder C. To prevent any spreading out of the electron stream, a
solenoid carrying a current was arranged coaxial with the tube. The

> Franck and Hertz, Verb. d. Deutsch. Phys. Ges., XV., p. 34, 1913.
t Goucher, Phys. Rbv., VIII., p. 561, Nov.. 1916.
* Bazzoni, Phil. Mag., XXXII., p. 566, Dec., 1916.

496 -A- •"'• BUGHES AND A, A. DIXON. ^SSl

electrons would be compelled to travel in narrow spirals along the lines
of magnetic force, and so to keep to the center of the tube. A small
field of a volt or two inside the cylinder was sufficient to drive the positive
ions produced to the electrode D (a strip of platinum about three mm.
wide). The advantage of this form of apparatus was that, once the
electrons had passed into the cylinder, their velocity would hardly be
affected by the small field inside the cylinder, and consequently they
would be available for ionization almost all along their path, in contrast

with the conditions obtained with the Franck
^^^^^^^^___ and Hertz type of apparatus. The curves

3^ " ' |£ \^ in Fig. 2 show the way in which the ioniza-

tion inside the cylinder varies with the po-
tential difference accelerating the electrons
from the disc to the cylinder. One curve
Fig. 1. was obtained with the Gaede mercury pump

running continuously, so that the residual gas
was mercury vapor at a pressure of about .002 mm. The second curve
— showing less ionization — ^was obtained with air also in the apparatus
at a pressure of .0026 mm. The intersection of these curves with the
axis takes place at about 9.45 volts. We must, however, take into account
the fact that the photo-electrons are emitted with a small velocity from
the disc. How much to allow for this velocity of emission is rather difficult
to say. Since the line X 2 ,537 is by far the strongest line in the ultra-violet
spectrum of the mercury arc, we have taken this to be, for our purpose,
the shortest line emitted by the mercury lamp. From Richardson's and
Compton's curves^ on the distribution of velocities among the photo-
electrons, it is evident that most of the electrons excited by the shorter
lines have velocities less than those of the fastest electrons produced by
^ 2,537. We therefore consider it justifiable to neglect the light of
shorter wave-length than X 2,537. From Richardson and Compton's
experiments we know that the long wave-length limit of the photo-
electric effect for platinum is X 2,910. By means of the equation

Ve = hv — hvo

we can get the velocity (measured in equivalent volts) of the fastest
electrons due to X 2,537. Taking hfe «= 4.13 X lO"^ (volt-frequency
units, Millikan) and

__ 3 X'lo^^ ^ 3 X 10^^

^ "" 2537 X io-« ' ^^ ~ 2910 X io-« '

we get V = .63 volt. Adding this to the accelerating potential 9.45 volts,

i Richardson and Compton, Phil. Mag., XXIV., p. 577« I9i3.

Nas. J

THE IONIZING POTENTIALS OF GASES.

497

we get 10.08 volts, or to the nearest tenth of a volt, lo.i volts, as the
ionizing potential of mercury vapor. (We may note that the straight
part of the curve intersects the axis at 10.15 volts. This could be
associated with the electrons emerging with practically no initial ve-
locity, though it can be shown that this agreement could only be expected
as an approximation.)

To find the ionizing potentials of other gases, the mercury vapor was
frozen out of the experimental tube by surrounding a U-tube between the
experimental tube and the pimip with carbon dioxide snow. It can
readily be shown, and verified by experiment, that there is a certain
pressure at which the ionization is a maximum, for if the pressure be too
low, there will be few molecules available for ionization, and if the pressure
be too high, the electrons will be used up before they enter the cylinder.

•4

"T-r

A. Hf vafurr omtLoJUL
B Ho va^ufr aJamt

Fig. 2.

Fig. 3.

On carrying out the experiments, using hydrogen, oxygen, and methane in
turn, and choosing the most favorable pressure, it was found that the
ionization was surprisingly small in comparison with that in mercury
vapor. When investigating the ionizing potential in mercury vapor, a
small negative current was obtained for accelerating potentials below the
ionizing potential. This did not cause any inconvenience, as the ioniza-
tion curves were so steep. Fig. 3 shows that the inclination of the
ionization curve for methane (the hydrogen and oxygen curves were
much the same) was much less than for mercury vapor, and moreover,
the negative part of the curve is greater in the absence of mercury vapor.
The negative part could not be reduced appreciably either by increasing
the magnetic field or by increasing the electric field inside the cylinder.
This made it impossible to determine the ionizing potential for gases
other than mercury vapor with any accuracy, as there was no definite
discontinuity in the curves. The method was therefore of little use, in
spite of its attractive features, except for the determination of the
ionizing potential of mercury vapor. It served to show that the ioniza-

498

A. LL. BUGHES AND A. A. DIXON.

tion of mercury vapor by electrons with energy exceeding lo.i volts is
much more intense than that of other gases even when the most favorable
pressures are selected. No evidence for ionization by electrons with
energies between 4.9 volts and 10 volts was obtained; had the electron
current been more intense, it would probably have been observed.

No satisfactory explanation of the negative part of the curves can be
given. It was not due to light getting into the cylinder. Some electrons
might possibly stick to molecules and become negative ions travelling with
the ordinary molecular velocities. For such velocities, however, the
electric field inside the cylinder should be ample to prevent any negative
ions from reaching the electrode. The electrons may rebound with
their full velocities from molecules. Even so, the magnetic field should
be sufficient to prevent them reaching the electrode D, If this Were the
explanation, it would suggest that the collisions with the mercury
molecules were non-elastic, while those with the other molecules were
elastic, a result not in agreement with Franck and Hertz's experiments.

(^•/«£i#mjCK

Fig. 4.

Second Method.

The method which was finally adopted for the measurement of ionizing
potentials was the same in principle as that of Franck and Hertz. A
diagram of the apparatus is shown in Fig. 4. It was made entirely o^

glass and platinum, the amount of metal
used being reduced to a minimum. Be-
tween what may be called the " experimen-
tal tube " and the pump on the one hand
and the supply bulb stopcock on the other,
traps were provided by which any vapor
could be frozen out by liquid air or some
other cooling agent. The electrons were
emitted from a filament F, about two or
three mm. away from the disc JD, which was
provided with a narrow slit 2 mm. X .5 mm. across the direction of the fila-
ment. These electrons were accelerated by an electric field between the
filament and the disc. About 5 mm. beyond the disc was an electrode E,
To get an ionization curve, E was connected to an electrometer (sensitivity
about 1,000 divisions per volt) and the filament F to a positive potential
of 3 volts. This prevented any electrons emitted by the filament F from
reaching the electrode E. The potential (positive) of D was varied so as
to increase the accelerating potential step by step, and the positive charge
acquired by E was measured against the accelerating potential. To get
a velocity distribution curve, the difference of potential between F and D

Vot.X.1
Na4. J

THE IONIZING POTENTIALS OF GASES,

499

was maintained at some suitable value, and the negative charge acquired
by E was measured for different retarding potentials between D and £.
Fig. 8 is a typical velocity distribution curve. It shows that the " 7.46
volt " electrons (electrons produced by an accelerating field of 7.46 volts)
have velocities corresponding to values between lo.o volts and 11.5
volts. (Strictly speaking, the real energy distribution curve, and from
it the real velocity distribution curve, is given on differentiating the
experimental curve in Fig. 8. As all the information we require can be
obtained at a glance from the experimental curves, it was thought unnec-
cessary to differentiate each curve.) When we used apparatus made of
glass and brass joined together with sealing wax, the velocity distribution
curves were generally very unlike those shown in this paper. The curves
seemed to indicate that electrons of all velocities from zero up to a maxi-
mum (often considerably less than that corresponding to the applied
potential) were present. This may be attributed to the formation of
polarization layers on the metal surfaces which have the property of
modifying the field in the apparatus very considerably. The state of
these polarization layers very probably changes rapidly on passing
through an aperture from one side of a plate which receives many elec-
trons to the other side which receives none. There will therefore be
strong electric fields in the neighborhood of the aperture which may
change the velocity and the direction of motion of the electrons passing
through the aperture. This experience leads us to doubt conclusions

— \ — r

BntoM fLnt (f-S yt4lt9

Fig. 5.

Hjx

taJiox

uons

f (

r—

Fig. 6.

drawn from experiments in which slow moving electrons are involved,
when the experimental apparatus contains brass or such metals joined
to glass by sealing wax and with tap grease in close proximity to the
place where the electrons are impinging on surfaces. Velocity distribution
curves should be taken, as it is unlikely that the actual velocity of the
electrons really corresponds to the applied potential. Indeed, with our
apparatus, designed to reduce surface polarization effects as much as

500

A. LL. HUGHES AND A. A, DIXON.

rSSCOKD

LScEm.

possible, we had distinct evidence that with some gases slight surface
films were formed.

When working with gases at a pressure of about .01 mm. of mercury,
it is difficult to be sure that the gas remains pure, especially if there is a
glowing filament in the apparatus. We therefore used the constant
flow method of supplying the gas. A supply bulb, of a liter capacity.

mm fttiinZtk/

Fig. 8.

was filled with gas prepared from pure chemicals and purified according
to the usual methods. Between this bulb and the experimental tube was
a very fine capillary tube, through which the gas flowed slowly. The
gas was removed from the apparatus by a Langmuir condensation pump.
By adjusting the pressure in the supply bulb, which determined the rate
of flow through the capillary tube, the pressure in the experimental tube
could be maintained at any value below .1 mm. Vapors were prevented
from entering the experimental tube by cooling the traps T and S on each
side of it by liquid air. In some cases, in which liquid air would reduce
the pressure of the gas under observation to practically zero, carbon
dioxide snow was used.

The pressure of the gas gradually diminished during the course of a
set of observations owing to a decrease in the amount of gas in the supply
bulb. The pressures at the beginning and at the end of a set of observa-
tions are indicated for each gas. To save space, the ionization curves
and the corresponding velocity distribution curves are shown for four
gases only. They are chosen so as to illustrate the greatest variation in
the shapes of the curves. These will be taken as types and the curves
for the other gases will be indicated by reference to one or other of these
four types.

An uncoated platinum filament was used to supply the electron current
in the experiments on the first gases worked with, viz., ethylene, methane,
ethane and carbon dioxide. A lime-coated filament was used in all the

Vol. X.1

THE IONIZING POTENTIALS OP GASES,

501

succeeding experiments. The electron currents were unusually small in
the gases carbon dioxide, oxygen, hydrochloric acid, chlorine and bromine,
and the filament had to be heated almost to its melting point. On
account of their chemical activity, the experiments on bromine and
chlorine were troublesome to carry out and consequently only one set of
observations was made for each of these gases.

The Results.

Mercury Vapor. — ^The ionizing potential of mercury vapor was found
by running the condensation pump until the pressure was below .00001
mm. (the limit of the gauge). As the traps were not cooled by liquid air,
the only gas present in appreciable quantity was mercury vapor. The
ionization curve for mercury vapor is shown in Fig. 5. The curve starts
from the axis at 9.5 volts; we shall refer to this point as the ** break
point." The velocity distribution curve obtained in mercury vapor is
shown in Fig. 6. The actual accelerating potential for the electrons was
9.66 volts, the velocity distribution curve shows that the fastest of them

^1 I I I

Fig. 9.

Fig. 10.

had a velocity corresponding to 10.35 volts. We therefore correct the
applied potentials in the ionization curves by adding 10.35 "" 9-^^ volts
to the value 9.5 volts which gives 10.19 volts as the ionizing potential
for mercury vapor. No evidence of ionization in mercury vapor by
electrons with velocities corresponding to 4.9 volts, or to any potential
between this and 10.2 volts, was obtained. The pressure of the mercury
vapor was not specially adjusted so as to give the maximum amount of
ionization. If ionization is produced by electrons with velocities below
10.2 volts, it is clear that it is of a different order from that produced by
electrons with velocities greater than this value. We believe, however,
that the explanation given by Bohr^ and by Van der Bijl,* that the ioniz-

1 Bohr, Phil. Mag., XXX.. p. 4x0. Sept., 1915.
•Van der Bijl, Phys. Rev., IX., p. 173, Feb., 1917.

502

A. LL. HUGHES AND A, A. DIXON.

ation produced by electrons whose velocities are below lo^ volts is a
secondary effect, and does not represent real ionization by impact, is
probably correct.

Hydrogen. — Hydrogen was prepared by the action of pure caustic
potash on pure aluminum. It was passed through a tube of red hot
copper gauze to remove traces of oxygen and then through a spiral
inunersed in liquid air. The supply bulb was filled several times with
hydrogen so prepared and pumped out completely between each filling.

It will be noticed that the ionization curve in hydrogen differs some-
what from that in mercury vapor. The velocity distribution curve A

I I I I

Fig. 11.

^ 6 i

Fig. 12.

was obtained immediately after the ionization curve A and the former
was used to deduce the correction to the ionizing potential as obtained
from the latter. The pressures given are those obtained at the beginning
and at the end of the set of observations.

A, Pressure .0303 — .0176 mm.
Break point 9.5 volts.

Correction from the velocity distribution curve, 11.56 — 10.86 volts.
Ionizing potential 9.5 + .7 = 10.2 volts.

B. Pressure .0152 — .0137 mm.
Break point 9.5 volts.

Correction from the velocity distribution curve, 11.56 — 10.86 volts.
Ionizing potential 9.5 + .7 = 10.2 volts.

To test whether the ionization curve was really due to hydrogen, the
flow of gas was stopped, and in a short while the pressure was down to
.00001 mm. At the same time, the ionization current for 14.46 volts
accelerating potential was reduced from loi to 5, showing that the ioniza-
tion curves were almost entirely due to hydrogen.

Oxygen. — Oxygen was prepared by the action of water on " Oxone "
cartridges. The oxygen was said to be 99.4 per cent. pure. It was
passed through a soda lime tube to dry it. Several liters were prepared

No'i^] ^^^ IONIZING POTENTIALS OP CASES. 503

and condensed in a tube surrounded by liquid air. About one third of
this was allowed to boil away, the next third was used to wash out the
supply bulb and finally to fill it, and the last third was rejected. There
is reason to believe that the oxygen actually used in the experiment was
very pure. The ionization curve for oxygen rises slowly from the axis
for several volts and then very quickly, as though the gas is much more
easily ionized by electrons whose velocity is a few volts above the ionizing
potential. Owing to the small electron currents and the correspondingly
small ionization currents, it was difficult to decide exactly where to place
the break point.

A. Pressure .0176 — .0160 mm.
Break point 9.3 volts.

Correction from the velocity distribution curve 9.76 — 9.82 volts.
Ionizing potential 9.3 — .06 = 9.24 volts.

B, Pressure .0116 — .0102 mm.
Break point 9.3 volts.

Correction from the velocity distribution curve 9.56 — 9.76 volts.
Ionizing potential 9.3 — .20 = 9.10 volts.

Hydrochloric Acid, — Hydrochloric acid gas was prepared by dropping
sulphuric acid on pure sodium chloride. The gas was condensed by
liquid air, and allowed to evaporate, the middle portion being taken to
wash out the supply bulb and to fill it with the gas to be tested. The
electron current from the filament was much smaller than usual. The
shape of the velocity distribution curve indicated that electrons of all
velocities were present or else that some electrical distribution around
the edge of the hole caused the electrons to deviate from their straight
line paths to a considerable extent. Such a state of affairs might [X)6sibly
be brought about by a small polarization layer on the surface of the
electrode D. The ionization curve is similar to that of oxygen.

A. Pressure .0116 — .0116 mm.
Break points 9.0 volts.

Correction from the velocity distribution curve 10.86 — 10.56 volts.
Ionizing potential 9.0 + .30 = 9.30 volts.

B. Pressure .0221 — .0212 mm.
Break point 9.50 volts.

Correction from the velocity distribution curve 10.76 — 10.56 volts.
Ionizing potential 9.50 + .20 = 9.70 volts.

Carbon Monoxide, — Carbon monoxide was prepared from formic acid
and concentrated sulphuric acid. The gas passed through caustic potash

504 A, LL. HUGHES AND A, A. DIXON, [toS2

solution and then through a spiral immersed in liquid air. The supply
bulb was filled and pumped out several times before the final filling.

Ionization curve: hydrogen type.

Velocity distribution curve: hydrochloric acid type.

A. Pressure .0212 — .0185 mm.
Break point 7.45 volts.

Correction from the velocity distribution curve, 7.10 — 7.52 volts.
Ionizing potential 7.45 — .42 «= 7.03 volts.

B, Pressure .0144 — .0130 mm.
Break point 7.45 volts.

Correction from the velocity distribution curve, 7.10 — 7.46 volts.
Ionizing potential 7.45 — .36 = 7.09 volts.
C Pressure .0116 — .0109 mm.
Break point 7.50 volts.

Correction from the velocity distribution curve, 7.32 — 7.42 volts.
Ionizing potential 7.50 — .10 = 7.40 volts.

Carbon Dioxide. — Carbon dioxide was prepared by heating sodium
bicarbonate. The gas was passed through concentrated sulphuric acid
and then solidified in a tube surrounded by liquid air. This was then
allowed to evaporate, the middle portion being used to wash out the
apparatus and to fill the supply bulb. Fractionating a gas from the solid
state instead of from the liquid state is probably less satisfactory as a
means of purification. The vapor traps in these experiments were cooled
by a mixture at — 90® C. instead of by liquid air.

Ionization curve: hydrogen type.

Velocity distribution curve: oxygen type.

A. Pressure .0137 — .0123 mm.
Break point 9.85 volts.

Correction from the velocity distribution curve, 10.2 — lo.o volts.
Ionizing potential 9.85 + .20 = 10.05 volts.

B. Pressure .0168 — .0168 mm.
Break point 9.70 volts.

Correction from the velocity distribution curve, 10.20 — 10.00 volts.
Ionizing potential 9.7 + .2 = 9.9 volts.

Nitrogen. — Nitrogen was prepared by heating sodium nitrite and
ammonium chloride with a little distilled water. The gas was passed
through concentrated sulphuric acid and through a tube containing red
hot copper gauze. Several liters of gas were generated and used to wash
out the apparatus before the final filling of the supply bulb was made.

Ionization curve: mercury vapor type.

Velocity distribution curve: oxygen type.

VQIUX.J p^^ IONIZING POTENTIALS OP GASES. 505

A. Pressure .0123 — .0116 mm.
Break point 7.45 volts.

Correction from the velocity distribution curve 7.72 — 7.52 volts.
Ionizing potential 7.45 + .20 = 7.65 volts.

B. Pressure .0102 — .0096 mm.
Break point 7.8 volts.

Correction from the velocity distribution curve 7.52 — 7.52 volts.
Ionizing potential 7.8 + o = 7.80 volts.

C. Pressure .0250 — .0221 mm.
Break point 7.9 volts.

Correction from the velocity distribution curve 7.32 — 7.52 volts.
Ionizing potential 7.9 — .2 = 7.70 volts.

Hydrogen Sulphide. — Hydrogen sulphide was prepared by the action
of dilute sulphuric acid on ferrous sulphate. The gas was washed through
dilute sulphuric acid and then frozen by liquid air. This was allowed to
liquefy and then to evaporate, the middle portion being used to wash out
the apparatus and to fill the supply bulb with the gas for the experiment.
As this gas liquefies easily, the traps were surrounded by a mixture at
about —70"* C.

Ionization curve: mercury vapor type.

Velocity distribution curve: hydrochloric acid type.

A. Pressure .0160 — .0152 mm.
Break point 9.2 volts.

Correction from the velocity distribution curve 10.40 — 10.16 volts.
Ionizing potential 9.20 + .24 = 9.44 volts.

B. Pressure .0123 — .0116 mm.
C Pressure .0109 — .0102 mm.

D. Pressure ? — .0032 mm.

Break point for 5, C, and D 8.5 volts.

Correction from the velocity distribution curves, 10.76 — 10.16 volts.
Ionizing potential 8.5 + .6 = 9.1 volts.
Nitric Oxide, — Nitric oxide was prepared by the action of nitric acid
on pure copper. The gas was passed through distilled water and caustic
soda. It was then liquefied and as usual the middle fraction of the
evaporating liquid was passed into the supply bulb.
Ionization curve: mercury vapor type.
Velocity distribution curve: hydrogen type.

A. Pressure .0176 — .0152 mm.
Break point 8.9 volts.

Correction from the velocity distribution curve, 10.10 — 9.62 volts.
Ionizing potential 8.9 + .48 = 9.38 volts.

506 A. LL, HUGHES AND A. A. DIXON. [ISS

B. Pressure .0130 — .0123 mm.
Break point 8.8 volts.

Correction from the velocity distribution curve, 10.10 — 9.56 volts.
Ionizing potential 8.8 + .54 = 9.34 volts.

C. Pressure .0109 — .0103 mm.
Break point 8.5 volts.

Correction from the velocity distribution curve, 10.30 — 9.56 volts.
Ionizing potential 8.50 + .74 = 9.24 volts.

Ethane, — Ethane was prepared by the action of ethyl iodide on alcohol
in the presence of a zinc copper couple. The gas was passed through
alcohol and then through concentrated sulphuric acid and was finally
liquefied by liquid air. About five c.c. of the liquid were obtained.
About a third of the gas was allowed to evaporate and the middle third
was used to wash out the apparatus and to provide the sample for the
experiment, the remainder being rejected.

Ionization curve: mercury vapor type.

Velocity distribution curve: oxygen type.

A. Pressure .0203 — .0176 mm.

Break point 8.60 volts.

Correction from the velocity distribution curve 9.92 — 8.52 volts.

Ionizing potential 8.60 + 1.40 = lo.o volts.
5. Pressure .0109 — .0096 mm.

Break point 8.55 volts.

Correction from the velocity distribution curve, 9.92 — 8.52 volts.

Ionizing potential 8.55 + 1.40 = 9.95 volts.

Methane. — Methane was prepared by heating a mixture of sodium
acetate and soda lime. The gas was passed through caustic soda solution
and through concentrated sulphuric acid and was then liquefied. This
was distilled, the middle portion being used to wash out the apparatus
and to fill the supply bulb.

Ionization curve: hydrogen t3^pe.

Velocity distribution curve: hydrochloric acid type.

A. Pressure .0260 — .0221 mm.
Break point 8.22 volts.

Correction from the velocity distribution curve, 11.30 — 9.92 volts.
Ionizing potential 8.22 + 1.38 = 9.60 volts.

B, Pressure .0109 — .0096 mm.
Break point 7.8 volts.

Correction from the vek>city distribution curve, 10.10 — 8.52 volts.
Ionizing potential 7.8 + 1.58 « 9.38 volts.

No'if *] ^^^ IONIZING POTENTIALS OF GASES. 507

C. Pressure .0185 — .0168 mm.
Break point 7.9 volts.

Correction from the velocity distribution curve, 10.10 — 8.52 volts.
Ionizing potential 7.9 + 1.58 = 9.48 volts.

Acetylene, — ^Acetylene was prepared from calcium carbide and water.
The gas was washed through caustic potash solution, a silver nitrate
solution, and concentrated sulphuric acid. It was then solidified by
liquid air. As in the case of carbon dioxide, fractional distillation from
a solid is not likely to result in so pure a product as from a liquid.

Ionization curve: mercury vapor type.

Velocity distribution curve: hydrochloric acid type.

A. Pressure .0230 — .0212 mm.
Break point 8.80 volts.

Correction from the velocity distribution curve, 12.20 — 10.72 volts.
Ionizing potential 8.80 + 1.48 =» 10.28 volts.

B. Pressure .0137 — .0130 mm.
Break point 8.4 volts.

Correction from the velocity distribution curve, 10.90 — 9.46 volts.
Ionizing potential 8.4 + i .44 = 9.84 volts.
C Pressure .0090 — ? mm.
Break point 8.30 volts.

Correction from the velocity distribution curve 10.90 — 9.46 volts.
Ionizing potential 8.30 + 1.44 = 9.74 volts.

Ethylene, — Ethylene was prepared from sulphuric acid and alcohol.
The gas was passed through a condenser immersed in ice, caustic potash
solution, and concentrated sulphuric acid. The gas was liquefied and
distilled. The first third was allowed to boil away, the next was used to
wash out the apparatus and fill the supply bulb, and the remainder was
rejected. The gas used was probably very pure.

Ionization curve : mercury vapor type.

Velocity distribution curve: hydrochloric acid type.

A. Pressure .0336 — .0271 mm.
Break point 8.40 volts.

Correction from the velocity distribution curve 9.65 — 8.22 volts.
Ionizing potential 8.40 + 1.43 = 9.83 volts.

B. Pressure .0144 — .0130 mm.
Break point 8.40 volts.

Correction from the velocity distribution curve 9.70 — 8.22 volts.
Ionizing potential 8.40 + 1.48 = 9.88 volts.

508 A, LL, HUGHES AND A, A, DIXON. ^SS

Chlorine. — Chlorine was prepared by the action of sulphuric acid on
potassium permanganate. It was liquefied by liquid air and distilled
as usual. The electron current was unusually small, making it difficult
to determine the exact position of the break point.

Ionization curve: oxygen type.

Velocity distribution curve: hydrochloric acid type.

Pressure

Break point 9.10 volts.

Correction from the velocity distribution curve 7.9 — 8.8 volts.

Ionizing potential 9.1 — .9 = 8.2 volts.

Bromine. — Bromine was introduced into a small tube in place of the
supply bulb. Its own vapor pressure was sufficient to drive enough
vapor through the capillary tube into the apparatus. As there was no
stopcock between the bromine tube and the experimental part of the
apparatus, it was unnecessary to cool the trap T. The bromine was
condensed in the trap S, causing a constant flow of bromine vapor through
the apparatus. The electron current was small in this experiment.

Ionization curve: mercury vapor type.

Velocity distribution curve: hydrochloric acid type.

Break point 10.5 volts.

Correction from the velocity distribution curve 8.1 — 8.6 volts.

Ionizing potential 10.5 — .5 = lo.o volt.

Sulphur. — ^The ionizing potential of sulphur was not looked for directly,
but in the course of the experiments, results were obtained which might
be used with some justification to calculate the ionizing potential of
sulphur vapor. In the experiment following that on hydrogen sulphide,
some anomalous results were obtained, for on reducing the pressure to
below .00001 mm. by the pump and preventing mercury vapor from
entering the experimental tube by liquid air, a large electron current was
still obtained. This result was quite contrary to what we had been led
to expect from our other experiments. From one ionization curve,. when
corrected as usual, a value of 8.44 volts was obtained for the ionizing
potential, and from another ionization curve a value of 8.24 volts was
obtained. That this is probably due to sulphur may be inferred from
the fact that there was no appreciable amount of gas or mercury vapor
in the experimental tube, and also that on heating the experimental tube
at the end of the observations some sulphur was driven out. This had
probably been the result of a partial decomposition of the hydrogen
sulphide by the hot filament. On cutting down the experimental tube
and cleaning out the sulphur by heating in a current of air, and repeating

VoL.X.1

THE IONIZING POTENTIALS OP GASES.

509

the experiments again, it was found that the source of the ionization
curve was removed and normal results were obtained. These values,
attributed to sulphur, hardly deserve as much credit as the other results.

Summary of Results.

The results are summarized in the following table, in which the values
of the ionizing potentials are given to the nearest tenth of a volt. One
can hardly claim an average accuracy of more than about .2 or .3 volt,
as it is difficult to say exactly where the ionization curve begins and where
the velocity distribution curve cuts the axis. (Theoretically, of course,
it never actually cuts the axis.) However, for purposes of comparison
between the different gases, one can probably claim an average accuracy
of about .2 volt.

Ionising PotenHals,

Gas.

This
Investiga-
tion.

Prmnck

and
Hertx.

Qoucher.

Basxoni.

Compton's
Theory.

Atomic
Volumes.

Atomic
Radii.

10.2
9.2
7.7
8.3?
8.2

10.0

10.2
9.5
7.2

10.0
9.3
9.5

10.0
9.9
9.9

20.5

16
12
11

7.5

10.25?
7.4

10.0

20.0

22.8

16.8
8.2

11.8
8.4
8.05
4.25
4.94

4.65
6.50

6.47

6.54

23.5
19.2
28.0
9.2
11.2
13.7
15.5
21.4
25.6
14.8

1.11X10-*

1.81

1.34
1.81
1.90

CI,

Br,

HCl

CO,

CH4

C,H.

C,H4

C,H,

2.68

1.88
2.28
1.86

2.75

The experimental results obtained by Franck and Hertz, by Goucher,
and by Bazzoni, are shown in the table. Where comparison is possible,
there is good agreement. The values predicted from Compton's theory'
are also given in the table. In the last two columns will be found the
atomic values taken from a paper by Harkins and Hall,* and the atomic
radii taken from Jeans's Dynamical Theory of Gases (2d edition, p. 341).

* Compton, Phys. Rev., VIH., p. 41a. Oct.. 1916.

* Harkins and Hall, Am. Cbem. Soc. Jour., XXXVIII., p. 169, Feb., 1916.

5 to a. ll. hughes and a. a, dixon.

Discussion of the Results.

Shape of the Ionization Curves. — ^The variations in the shape of the
ionization curves are greater than can be accounted for by differences in
the velocity distribution curves. This implies that the way in which
the ionization by collision depends upon the velocity of the electron
(above the ionizing potential) differs for different cases. If we assume
that the ionization per collision is constant for velocities above the
ionizing potentials then it can be shown that the straight part of the
curve, when produced, intersects the axis at a point which gives approxi-
mately the ionizing potential, provided we refer this point to the voltage
corresponding to the most probable energy of the electron, and not as
we have done hitherto, to the energy of the fastest electrons. Such is
the case for ethylene, and approximately so for mercury vapor, but not
at all the case for oxygen. (Some of the curves have no straight por-
tions.) As these experiments were not designed primarily to investigate
this relation, the matter will be left without further comment.

Ionizing Potentials of the Elements. — ^A knowledge of the ionizing poten-
tials of the elements should furnish material for testing theories of atomic
structure. The only theory which is sufficiently well developed to enable
us to make a quantitative comparison is that due to Bohr,^ and even
Bohr's theory is only worked out in sufficient detail to allow us to test
the results for hydrogen and helium.

Bohr pictures a hydrogen atom as one electron rotating about a positive
nucleus possessing one unit positive charge. The only orbits which are
possible are those in which the electron has angular momentum equal to
some integral multiple of A/^t. The negative energy of the atom is

2Tm^ I I

/I* T r

where r is the number of the orbit, being i for the innermost orbit, 2 for
the next, and so on, and K is Rydberg's constant, 3.29 X 10^*. This
will give the energy required to remove the electron from the orbit to
infinity. The work required to remove the electron from the wth orbit
to the nth is

\m» n^)'

For purposes of comparison with experiments, it is convenient to express
the work in terms of equivalent volts. On this basis, K corresponds to
13.5 volts, and this measures the energy required to remove the electron

> Bohr, Phil. Mag.. XXVI.. p. 857, Nov., 1913.

No"^*] ^^^ IONIZING POTENTIALS OP GASES. 5 I I

from the innermost ring. We should expect this to represent the energy
necessary to ionize the normal atom. The work required to move the
electron from the first to the second ring would be

Kl-z r I , or lo.l volts.

This is very close to the ionizing potential of hydrogen as found by
experiment, but does not, on the other hand, represent the work required
to remove an electron completely, which we should naturally consider
to be the work of ionization.

It might be argued that we are working with the hydrogen molecule
and not with the hydrogen atom. According to Bohr, the molecule may
be pictured as two positive nuclei with two electrons rotating about the
line joining them. E^ch electron has angular momentum equal to some
multiple of the constant of angular momentum. The molecule in its
normal state has unit angular momentum for its electrons. The negative
energy of such a molecule is 2.20K and the negative energy of such a
molecule with one electron completely removed is .SSK. The work
required to remove the electron is therefore

2.20K — .SSK = 1.32X, or 17.7 volts.

Bohr however shows that the positively charged molecule is unstable,
and prefers to regard the ionization of the hydrogen molecule as the dis-
integration of the molecule into a single nucleus and a hydrogen atom.
This requires energy of the amount

2.20K — jK = 1.20K, or 16.2 volts.

Bohr pictures the normal helium atom as two electrons rotating round
a doubly charged positive nucleus, each electron having angular momen-
tum equal to some multiple of the unit angular momentiun. The
negative energy of the normal helium atom, the electrons being in the
innermost ring, is 6.13K, and the negative energy of the positively charged
helium atom, with one electron in its innermost orbit, is 4K. The work
to remove one electron is therefore

S.iiK — 4iC = 2.iiKt or 28.6 volts.

The work to remove one electron, not to infinity, but to the next orbit,
will be some fraction of this. Its value will depend upon the assump-
tions as to the rearrangements of the orbits; we may probably assume
that it will not be very different from 3/4, the value previously obtained

512 A, LL. HUGHES AND A. A. DIXON. [i

for the hydrogen atom. This gives 21 volts. It is significant that the
ionizing potentials of hydrogen and helium do not agree with the values
calculated for the complete removal of an electron from the respective
atoms, but do seem to agree with the values of the energy required to
move an electron from the innermost ring to the next. Moreover, the
experiments of Bazzoni show no traces of any extra ionization setting in
at 28.6 volts, nor do our experiments show any discontinuity in the curves
for hydrogen at 13.5 volts or at 16.2 volts. It may be inferred that the
ionization does not consist of the direct expulsion of an electron from an
atom by the impact of another electron, but, as Bohr has suggested in
the case of helium, is the result of a transition from the normal state of
the atom to the next stationary state. This may come about in several
ways. To test whether ionization consists in the transfer of an electron
from the innermost orbit to the next, followed by a complete removal
from that orbit by a second collision with an electron, we measured the
ratio of the ionization currents to the electron currents. If we assume
that atoms with electrons displaced to the second orbit tend to go back
to the normal state, then, if the electron current is small, almost all the
atoms will go back to their normal state before they are hit a second
time, while if the electron current be large, there would be a much
greater chance for an atom to be struck a second time before it had got
back to its normal state. Our experiments showed that the ionization
current was doubled when the electron current was doubled, and therefore
this linear relation gives no support to this particular view of the me-
chanism of ionization.

For gases other than hydrogen and helium, we must do without so
definite a theory and content ourselves with searching for general re-
lations. There is no clear connection between the ionizing potential of
an element and its electronegative or electropositive character. On
Bohr's theory we have seen that the closer the electron is to the nucleus,
the more is the energy required to take it away. We might therefore
try the effect of comparing the ionizing potential with the radius of the
atom. Provided we keep to elements which are not too widely separated
in character, an increase in atomic volume and atomic radius is accom-
panied by a decrease in the ionizing potential. This is the case for the
inert gases, if we disregard neon, as its atomic volume is not so well
established as that of the other gases. It is also the case for hydrogen,
oxygen, and nitrogen. It is not the case however for bromine and
chlorine. It is well to remember that the experimental difiiculties were
greater in the case of these gases than in the case of other gases. This
relation is similar to one noticed by one of the authors,^ viz., that there is

^Hughes. Phil. Trans., CCXII., p. 205, 1912,

Na"s^*l ^^^ IONIZING POTENTIALS OP GASES. 5 1 3

a regular decrease in the energy required to detach a photo-electron from
a metal as the atomic volume increases. This relation holds only for
elements within the same column of the periodic table; there is a discon-
tinuity (always in the same direction, however) as we pass from one
column to the next. A close relation between the atomic radius and the
ionizing potential for all the elements could hardly be expected, as the
arrangement of the electrons in the atom must be a factor in determining
the energy required to detach an outermost electron in addition to the
radii of their orbits. According to Ludlam^ chlorine is not ionized by
the ultra-violet light which is capable of ionizing air. This is in agree-
ment with these experiments if the ionization of air by ultra-violet light
is due to the ionization of nitrogen alone, but not so if the oxygen is
ionized as well.

Compton* recently proposed a theory from which he deduced a relation
between the ionizing potential V and the specific inductive capacity K
of a gas.

.194

7 =

^K-

This formula agrees fairly well for most of the elements; the agreement is
poor however for mercury, chlorine, and sulphur, if our value be correct.

Ionizing Potentials for Compounds. — ^The results do not point to any
definite relations for compounds. The ionizing potential for hydrochloric
acid is intermediate between that of hydrogen and that of chlorine. The
same relation holds for hydrogen disulphide, hydrogen, and sulphur, but
does not for oxygen, nitrogen, and nitric oxide. In working on the photo-
electric effect of solid compounds' it was noticed that, in general, the more
stable the compound (measured by its heat of formation) the less was its
photo-electric effect and presumably the more difficult it was to detach
an electron. It might be reasonably expected therefore that the ionizing
potentials would be greater for the more stable compound gases. On
testing this out no sort of agreement could be found. Indeed, the
ionizing potential is almost constant for the four hydrocarbons, two of
which are exothermic and two endothermic.

The values of the ionizing potentials calculated from Compton's
theory, are little more than half the experimental values. Compton
does not expect a good agreement on account of the uncertainty in the
values of the specific inductive capacity K for the compound gases.

> Ludlam, Phil. Mag., XXIII., p. 757, 1912.

* Compton, Phys. Rbv., VIII., p. 412, Oct., 1916.

* Hughes, Phil. Mag., XXIV., p. 380. 191 3.

514 a. ll. hughes and a. a, dixon, [sss

Summary.

The ionizing potentials of fifteen gases have been measured by a
method similar in principle to that of Franck and Hertz. A second
method of measuring the ionizing potential was worked out, but gave
satisfactory results only in the case of mercury vapor.

The ionization of mercury vapor by electrons whose energy is some-
what greater than that corresponding to the ionizing potential is much
more intense than is the case for the other gases.

Several relations which might be expected to account for the values of
the ionizing potentials have been suggested. The experimental values,
however, do not agree well with any of them.

No*/^'] ^^^ PLANCK RADIATION CONSTANT Cs. 5 I5

A DETERMINATION OF THE PLANCK RADIATION

CONSTANT Ct.

By C. E. Msndenhall.

THE cross-connections of Planck's radiation theory give an importance
and interest to its constants which is sufficient excuse for devoting
a considerable amount of time to their accurate determination. The
work to be described in the present paper involves some novelties of
method and conditions and some refinements of observation which it is
hoped have led to an increase in final accuracy. The work will be
discussed in two parts, in both of which however, the same method was
used, namely, that involving the measurement of the ratio of the intensity
of emission for a known wave-length at two known temperatures. For
the wave-lengths and temperatures involved the Planck equation

E = ci\-^ ^

is equivalent to the Wien form

to less than one part in ten thousand. Using the Wien form, the ratio
of intensities is given by

^^M£^) = y(7;^iT;)-

In the first part of the work, a graphite tube furnace was used, and
Ti (2705** K.) was determined from Ti (1604** K.) by the use of the Stefan
Boltzman law. The general arrangement of the furnace and some of its
attachments are shown in a previous paper.^ The methods used in
determining the observed quantities, X, Ei/Et, Ti and Ti will first be
considered.

1. The wave-length used was determined by an ocular slit in the focal
plane of a Hilger constant deviation spectroscope. The slit subtended
about 20 Angstr6ms, and its center was found by careful calibration, to
be at 6501 A.U.

2. The ratio Ei/Ef. In front of the collimator slit of this instrument

» Mendenhall and Forsythe, Phys. Rev., ad series, Vol. IV., p. 65, July, 1914.

5 I 6 C. £. MENDENHALL. [sam

was mounted a pair of achromatic lenses and a comparison lamp.
The first lens served to form an image of the black body dia-
phragm in the furnace, in the plane of the lamp filament, while the
second formed an image of this image and of the filament upon the
spectroscope slit. The arrangement is thus a spectro-optical pyrometer,
the comparison field consisting of a narrow band in the center from the
comparison filament, bordered above and below by light from the furnace.
By careful adjustment one very sharp separating line was obtained,
permitting adjustments of very considerable accuracy. The brightness
of the comparison filament was determined by reading the current
flowing through it, on a Siemens & Halske millivoltmeter provided with
such a shunt as to give nearly a full scale deflection, and read with a mag-
nifying glass. In this way the current necessary to give photometric
balance against the furnace at the temperature Ti was determined.
With the furnace at the high temperature Tt a large rotating sector (12
inches in diameter) was inserted in front of the first lens so as to reduce
the intensity of the light from the furnace. This disc had two apertures,
either of which could be closed by pasting over a piece of black paper.
One aperture reduced the intensity of the furnace light to a value slightly
greater, the other to a value slightly less than the initial intensity at
temperature Ti. One aperture had an opening of i** 17' 05'', the other
an opening of i® 12' 18". Thus by determining the lamp current required
to give photometric balance with each of these apertures in turn, it was
possible by linear interpolation, plotting lamp current against the log-
arithm of the sector reduction factors, to determine the aperture which
would have reduced the E^^^ exactly to the JExn- The quotient of 2t
by this aperture (in radius) is the ratio Et/Ei. The value of the sector
apertures was determined by very careful measurements on a Geneva
Society spectrometer. The losses at the glass window W, provided they
remain constant at Ti and Tj, obviously do not enter into this matter.
The linear interpolation referred to above between the two sectors was
shown to be allowable by determining the intensity of emission / of the
comparison lamp as a function of current, and plotting log / against
current. The resulting curve had such a slight curvature that for the
small range between the two sector apertures straight-line interpolation
was quite sufiicient.

3. The temperature Ti, This temperature was defined and repro-
duced as that having 14.91 times the intensity of radiation of a black body
at the melting point of gold, for the complex of wave-lengths transmitted
by two thicknesses of the standard red pyrometer glass No. 2745. This
temperature was chosen because it gave an intensity of light permitting

Nc^s^'l ^^^ PLANCK RADIATION CONSTANT Cs. 5 I 7

about the maximum accuracy of photometric balancing, with the spec-
troscopic outfit described above. It was practically realized by cali-
brating a pyrometer at the gold point using a platinum-wound black-body
furnace and determining the melting point by the wire method. This
pyrometer was then put in front of the graphite tube furnace, in place
of the spectro-pyrometer, a sector with a transmission ratio 1/14.91
rotated in front of it, and the furnace temperature raised until the
pyrometer indicated the correct gold point, the losses due to the glass
window W having been allowed for in the calibration. At the same time
a control pyrometer was sighted into the back of the furnace, and its
reading obtained when the proper temperature was reached as shown by
the front pyrometer. Of course this transfer or comparison was carried
through several times in connection with each ** determination " of d,
and before and during the series two very consistent determinations of
the gold point with the " front " pyrometer were carried out. After the
temperature Ti had thus been transferred to the back or control pyro-
meter, the front pyrometer was lifted away and replaced by the spectro-
scope.^ The actual determination of the value of Ti defined as above
was accomplished some time later, using a standardized thermo-couple
borrowed from the geophysical laboratory through the kindness of Dr.
Day. For the constants of this particular couple and for very valuable
suggestions as to the proper use of it, I am under obligations to Dr. W. P.
White. The same pyrometer and lamp was of course used, with two
different black body furnaces, and many determinations of the gold
point were made in this connection. As this determination of Ti is of
fundamental importance, and as it is not altogether simple even granted
a standardized couple, it must be considered a little more in detail. The
first furnace used, double wound with platinum on Marquardt porcelain
tubes, was almost exactly the same length as that in which Dr. White
had determined the constants of the couple, and the couple gave almost
exactly its standard E.M.F. at the gold point. The remaining de-
screpancy might be due to a difference in the furnace gradient, a difference
between the wire and crucible methods of determining melting points,
a difference in the purity of the gold, or a difference between the two
potentiometer and standard cell combinations. The last two factors
were eliminated by using a little gold wire from the geophysical laboratory,
which gave the same melting point as ours, and by measuring on our
potentiometer the E.M.F. of a number of copper-constantin couples

^ It may be asked why the " control '* or back pyrometer was not itself used to determine the
initial temperature; but it is evident that it is the temperature of the front face of the interior
graphite diaphragm that is desired, since it is at this face that the spectroscope and the Stefan
Boltzman apparatus both point.

5 1 8 C. E. MENDENHALL. [^S

arranged to have one junction in melting ice and the other in steam.
This is a standard arrangement of Dr. White's for comparing potentiom-
eters. My values for the E.M.F. of this ** tester " indicated that my
potentiometer read about 1/5500 high. As there was already consider-
able evidence that the wire and crucible methods give consistent values
for the melting point, this matter was not gone into, but it was concluded
that the difference between our value and the standard E.M.F. of the
couple at the gold point was due to a difference in furnace gradient.
This difference in E.M.F. may conveniently be referred to as the " fur-
nace correction." With the first furnace it amounted to only about 2/af
(.15** C), and it was assumed not to vary with temperature from the
gold point up to Ti. Some justification for this assumption will be re-
ferred to later. The optical pyrometer used in the Ct observations was
now used to determine a temperature of this furnace for which the in-
tensity of radiation through the double red filter (No. 2745) was 14.91
times that at the gold point, and the E.M.F. of the standard couple at
this temperature measured. The " furnace correction " was applied
to this to reduce to standard conditions — ^and from this the temperature
was determined. In this way five independent determinations of Ti
were made with this furnace as follows: 1332**.!, I33i**.9, I330**.9, 1330^1,
1330*^.2, mean 1331^.0 C. A longer furnace was next used, and the hot
end of the couple extended by wire taken from the other end so as to
bring the temperature gradient as nearly as possible along the same part
of the wires. The standard conditions were not so well reproduced in
this furnace as in the other, as the " furnace correction " was now about
lO/jLv (.8^ C). The windings of this furnace were in bad condition,
however, as was evident from the variations in the determinations of the
gold point, and only three determinations were made before it was
necessary to repair it. They were not as consistent as the first, and the
mean is considerably lower, 1328*^.9 C. Owing to the bad condition of
the furnace, and to the fact that the furnace couple was found, upon
taking apart for repairs, to have been misconnected, very little weight is
given these results in the final average. While repairing this furnace
opportunity was taken still further to improve it by lining the innermost
** black body " chamber with platinum of sufficient thickness to increase
the uniformity of temperature. The " black body " chamber of this
furnace was about 4 cm. long, and 1.5 cm. in diameter, with an aperture
about 4 mm. in diameter. As indicated by the lack of any optical de-
finition in the interior, and small difference in brightness between the
aperture and the diaphragm containing the aperture, the black body
conditions must have been very good indeed, as would be expected since

Nol^sf*] THE PLANCK RADIATION CONSTANT Ct. 5I9

the aperture was less than i per cent, of the internal radiating area.
There was no difference between the radiation measurements in the two
furnaces at the gold point. With the repaired furnace, which gave
beautifully consistent values for the gold point, three more determina-
tions of Ti were made, giving i332**.o, 1330^.8, I330**.9 C. Giving these,
for the reasons mentioned above, very considerably more weight than the
preceding three, the mean for the long furnace is 1330^.6 C, and the
close agreement of this with the mes^i for the short furnace is considered
to be some justification for the assumption that the ** furnace correction "
is independent of temperature. Finally the mean value obtained for
Ti is 1330*^.8 C, or with all the accuracy that may be claimed here,
1331** C, which is believed to be correct to 0^.5 C.

4. Tj. As has been previously described, the front part of the graphite
tube furnace chamber contained a thermopile, consisting of a single
junction with a thin blackened silver receiving surface, 3 nun. in diameter
at the center of a small hemispherical silver mirror, to increase the effective
absorbing power. This was connected to a suitable D'Arsonval galva-
nometer so that when exposed to the total radiation from the furnace at
temperature Ti a deflection of from 30 cm. to 40 cm. would be obtained.
By means of the control pyrometer and a suitable choke coil in the pri-
mary of the transformer supplying the furnace, it was possible to hold
the temperature constant while these thermopile readings were being
obtained. In order to determine the temperature Tj, a rotating sector
of suitable aperture could be swung into place in the furnace chamber
between the pile and the graphite black body furnace, and the tem-
perature of the furnace raised until upon exposing the pile, the same
deflection as before was obtained. If S is the transmission coefHcient
(i. e., ratio of angular opening to 2t), then according to the Stefan
Boltzman law we have Tt = Ti^li/S. In the meantime a suitable
rotating sector would be mounted in front of the control pyrometer, so
that a reading of the temperature of the back of the furnace could be
obtained, corresponding to Tt at the front. A somewhat fuller descrip-
tion of this part of the apparatus, designed for the measurement of tem-
peratures by the Stefan Boltzman law, will be found in a previous article.
It need only be mentioned here that the requisite diaphragms and shutters
were water cooled, and the sectors inside B were made double so as to
avoid any danger of their heating up and re-radiating to the thermopile.
A steady flow of nitrogen at about 12 mm. pressure was maintained
through the furnace chamber, and it has been found by previous tests
that the absorption of such an atmosphere was quite negligible. It
bhould be mentioned that the tests for absorption were made with

520 C. E, MENDENHALL, [iS^

furnace gases drawn out into a side tube and hence cold; but I know of
nothing to indicate that the absorption would be greater when hot. The
succession of operations incident to a single determination of Ct may now
be summarized as follows: First, the standard pyrometer was directed
into the front of the furnace, with a 14.91 sector rotating in front of it.
The furnace temperature was raised until the pyrometer indicated the
gold point. The spectroscope and attachments were then substituted
for the pyrometer, and the lamp current necessary to balance against
the furnace (with no sector) was carefully determined. Then the gal-
vanometer deflection (about 350 mm.) due to the total radiation ther-
mopile (with no sector) was measured. The temperature of the furnace
was then raised until the same deflection was obtained with the 1/8.33 1
sector rotating in front of the thermopile. Then the large sector was
rotated in front of the spectroscope, and the lamp current determined,
which would give photometric balance with each of its two apertures
exposed in turn. In two instances it was not possible to obtain satis-
factory observations with both apertures, and for these cases the slope of
the log /, current curve was taken from the work of the preceding and
following days.

Results. — ^Six series of this sort were carried out, each giving a value
of log £2/^1 as follows:

2468 2.469

2.464 2.466 Mean, 2.4663.^

2.466 2.465

The agreement may be considered extremely satisfactory, the extreme
range corresponding to values of Ct of 14,381 and 14,410. Gathering
together the data for the determination of Ct we have, corresponding to
the expression c% = log RiXTiTi/T^ — Ti), log R = 2.4663, X = 6,501,
Ti = 1,604, Ti = 2,725, from which d = 14,394.

Sources of Error, — (a) Wave-length. The error in the determination
of the wave-length corresponding to the center of the ocular slit is neg-
ligible, certainly not more than one Angstrom, but there is always the
danger of scattered light of shorter wave-length to which the eye is more
sensitive. The high dispersion of the Hilger prism (/ix> = 1.74) the
narrow slits used, and the relatively long interval from the region of
maximum eye sensibility to the wave-length used all tend to minimize this
source of error.

> In addition an earlier set in which the temperatures were measured through the back or
control end of the furnace gave log R » 3.467.

Vlo^S^] ^^^ PLANCK RADIATION CONSTANT Ct. 521

(i) In 7*1. Since an error of an entire degree in T would only produce
an error of i part in i,6oo in d, no very great error is to be expected from
this source, unless it arises from some change in the pyrometer occurring
between its use in the main part of the work, and the determination of
7*1 by thermocouple comparisons. And since the same red glass eye
screens were used in both cases, it is difficult to see how even a change in
the pyrometer could seriously affect the result. Further evidence of the
accuracy of the determination of Ti is given by the application of Wien's
law to the red glass optical pyrometer between the gold point and Ti,
Using for the effective wave-length of transmission of the double thickness
of glass No. 2745, as determined by Hyde Cady & Forsythe, X = .6671/i
for d the value above given, and for the ratio of intensities the measured
sector ratio 14,908, the resulting value of Ti is I33i**.4 C. This is
certainly as close an agreement as could be expected.

(c) Tt. There are here several possible sources of error. First, the
fact that the thermopile receiving surface was not perfectly absorbing
would enter only as a second order error, due to a possible difference in
the effective absorbing power at the two temperatures Ti and Ti resulting
from the change in the spectral energy distribution. No attempt was
made to determine the magnitude of this error, but as the maximum of
energy in the black body spectrum shifts only from about i.8/i to i.Oai
for these two temperatures and as the absorbing power of acetyline black
is almost independent of wave-length in this region, it seems safe to con-
clude that the error in question would be very small. The absorption of
the gases in the furnace chamber (12 nmi. pressure) would also produce
only a differential error, and as before stated it had been found that the
total absorption of these gases when cold, was certainly not more than
0.1 per cent. The sectors used in front of the thermopile were of large
aperture and could be readily measured with the necessary accuracy.
The galvanometer deflections of about 350 mm. could be read to o.i mm.
and they were so consistent and reproduceable as to give one great
confidence in their reliability. The question of galvanometer pro-
portionality did not enter, as the same deflection was used at Ti and Ti.

(d) R, the ratio of emissivities. In this connection the most difficult
factors to determine were the two sector apertures of 1.284, and 1.205.
In measuring these the smallest angle which could be read, by estimation,
was 5''. As each aperture was measured five times and each measure-
ment read on two verniers it would seem safe to expect an accuracy of at
least one part in a thousand in the final value, especially in the average
of the two, which is really what counts.

Summing up, it does not seem unreasonable to hope for an accuracy
of one part in five hundred in the final result.

522 c, e. mendenhall.

The Second Method.

This was exactly the same in principle, but applied directly between
the melting points of gold and palladium as determined by the wire
method. This simplifies the temperature measurements, but the tem-
perature range is so much less that the temperatures must be known with
greater accuracy, and the intensity of emission is so low at the gold
point that photometric balancing is far less accurate. In fact this work
was carried out merely as a check on that already described, to show
that the value of Ci obtained above was consistent with observations at
the lower temperatures. The various elements of the problem will be
briefly discussed.

The Furnace. — ^This was the long one referred to above, with a platinum
lining on all walls of the radiating chamber but the rear one at which the
pyrometers were sighted. Its extreme length was 70 cm., and the tem-
perature gradients were so small in the central chamber that five gold
melts would as a rule all fall inside an interval of .6^ C. and five palladium
melts in an interval of 2®.o C. Only the purest Hereans gold and
palladium were used, and melting points were determined at short
intervals, so that the changes in the thermocouples which occurred at
every heating to the palladium point (1549** C.) should be as much as
possible eliminated. The form of spectro-pyrometer previously used was
remodeled and improved for this work. The most serious source of error
is the optical defects of the comparison lamp bulb which diffuse the
dividing line between the two fields. A strip of thin platinum in air gave
much more accurate balancing conditions, but was not sufficiently
permanent. It is hoped to make improvements in this feature. The
ratio of intensities was again determined by interpolation between
sectors, on the log / current curve. Two different wave-lengths were
used, 0.5780A1 and 0.5460/i, chosen at the yellow and green mercury lines
so that the wave-length scale could be quickly checked. The collimator
slit covered about 20 Angstroms and the ocular slit about 20.

Results — ^The values of C% cover a considerable range, as was ex-
pected, and hence a considerable number, 17, separate determinations
were made. Aside from the wave-length other conditions such as aper-
ture of image-forming beam, and width of collimator slit were varied,
without producing much systematic change in the result. The individual
values are as follows: 14450, 14416, 14400, 14460, 14450, 14300, 14450,
14450, 14206, 14455, 14411, 14506, 14453, 14506, 14455, 14355, 14291-

While these results vary in a disappointing way, still the mean 14,413
is in about as close agreement as could be expected with the value found
under the very different conditions first described. Comparing the

No's^i ^^^ PLANCK RADIATION CONSTANT Cs. 523

values of d obtained with X =» .667/i, .577;* and .546/i, there is slight
evidence of a dependence of the value obtained upon the wave-length
used, but considering the uncertainty of the second method it is not wise
to attach any significance to this. As a final value, less weight being
given to the second series of observations, the round number 14,400 is
perhaps satisfactory.

Recent values to be compared with this are:

Reichsanstalt^. I4i300,

Coblentz* 14,322 and 14,369,

Hyde, Cady & Forsythe* 14,460.

Of these the latter is subject to the added uncertainty of having been
determined directly with red glass screens which are very far from
monochromatic. The present value falls in the middle of the range of
recent measurements, but differs from the results of the other most direct
determinations by much more than the supposed limit of error of any of
them.

The Relation of Ct to Other Constants.

According to Planck's theory Ct is related to a number of other
constants by the following equations:

, ^ C2 aCt' aCl

^^ "■ 4.9651 ' "" 4^TaC • '^ " 48ira'
where

a = 4<r/c = constant of total radiation.
h = Planck's quantum constant.
k = Planck's probability constant.
a = 1.0823.
c = vel. of light.

These equations give opportunity for three cross-connections or
checks, of which only the last two need be considered. It is of most
interest to compare the values of h and k determined by purely radiation
observations with the value of h determined by Millikan from photo-
electric methods, and the value of k computed by Millikan from his value
of e, the electric charge, and the gas constant R. In order to do this I
have used Westphall's latest value for a as quoted by Millikan namely,
5.67 X 10-" (watts/cm.* Deg.*)

» Mueller, Warburg et al.. Ann. Phys.. 48. 191S. 430.

* Coblentz. Phys. Rev.. 7» 1916.

• Hyde. Cady and Forsythe. Ap. J., June, 1915.

524 C. E, MENDENHALL. [^S

Using C% = 14400 the resulting values are:

h = 6.654 X lo-*',

* = 1.383 X io-i«.

Whereas Millikan's values are:

h = 6.547 X lo-*',!

* = 1.372 X 10-",*.

On the other hand, combining Millikan's value of k with the present
Cj, gives h = 6.585 X lo"*', which is in better agreement with Millikan's
photo-electric value.

If Coblentz's value of <r, 5.72 X io~" be used, discrepancy between
the radiation and the electrical values of h and k become still larger.
The contrast is put in the reverse way by Millikan, who computes from
electrical and gas data the values Ct = I4»3i2 and <r = 5.72 X lo"".
It will be observed that this value of d is well at the extreme of the range
of recent direct experimental results, while the value for <r agrees exactiy
with Coblentz and very closely with Westphall.

It remains for further work to decide as to the significance of these
discrepancies.

Dbpartmsnt op Physics,

University of Wisconsin,
May, 191 7.

1 Millikan. Proc. Nat. Acad. Sd., April, 191 7.
* Millikan, Phys. Rev.. 2, 1913. pp. 109-143.

VouX.
Na

J. ] SPECIFIC HEATS OP HYDROGEN. 525

A DETERMINATION OF THE RATIO OF THE SPECIFIC
HEATS OF HYDROGEN AT i8^ C. AND - 190** C.

By Margarbt Caldbrwood Shields.

T^HE methcxi for determining the ratio of the specific heats of gases
-*- originally presented by Lummer and Pringsheim* in 1898, and
since used in modified form by Moody* and by Partington,' is generally
conceded to be the most precise method thus far available, its only dis-
advantage being that it has seemed to require large quantities of gas.
Three years ago, however, H. N. Mercer* obtained with the use of sur-
prisingly small flasks some preliminary data which pointed to the possi-
bility of using the method with small scale apparatus. Accordingly
Professor Millikan suggested to the author two problems which were
obviously waiting such an opportunity.

First: There are in the literature of the subject at present only two
satisfactory determinations of the ratio for hydrogen at 20** C; 1.4084,
a direct determination by Lummer and Pringhseim, and 1.407, computed
by Scheel and Heuse* from their observations on Cp by the "constant
flow" method. Inasmuch as the kinetic theory affords no explanation
of values so high, and experimental data generally are now under close
examination from the point of view of the quantum theory, a careful
redetermination of the ratio is needed to decide whether a quantum
effect is actually manifested in hydrogen at this temperature.

Second: Eucken,* from observations on C», and Scheel and Heuse from
observations on Cp, have announced that the hydrogen molecule loses
almost entirely its two degrees of rotational freedom by the time the
temperature reaches — 180** C, and becomes virtually a monatomic gas,
the ratio of the specific heats being according to Eucken 1.604, ^^^
according to Scheel and Heuse 1.595. Will it be possible to confirm
this by direct observation of the ratio?

This paper is an attempt to answer these two questions.

> Ann. d. Phys.. 64: 536, 1898.
« Phys. Rbv., 34: 275, 1912.

• Phys. Zelt.. 14: 969. 1913.

' Pro. Roy. Soc. London, 26: 155, 1914.

• Ann. d. Phsrs., 40: 473, 1913.

• Ber. d. Preus. Akad., 1913, p. 141.

526 MARGARET CALDERWOOD SHIELDS. ^SS

I. At i8^.

Experimental Arrangements. — ^The method employed for this investiga-
tion is, as stated above, essentially that of Lummer and Pringsheim.
It consists in measuring the cooling attendant upon an adiabatic expan-
sion from pi to Pt\ the two pressures and temperatures are connected
for the ideal gas by the relation,

(W - (f:)'

and y is therefore obtained from the equation,

^" log"/,. -log"/..' ^^

The two modifications of the original experiment are (i) the use of a one
liter flask in place of a large carboy, and (2) the substitution of a minute
thermoj unction for the platinum resistance thermometer, following in
this respect the method already used by Moody in the Ryerson labora-
tory.

The thermal element was of .001 inch copper and constaiitan wires.
It was introduced into the flask by means of two glass tubes through the
rubber stopper, which were inside drawn out to fine capillaries and
bent into a Y, which spread nearly to the diameter of the bulb, and
could be folded together for insertion into the flask by twisting each of
the tubes through 90**. It was found by repeated effort that normal
values of y could be obtained only when the couple was thus introduced
with a minimum of glass as remote as possible from the junction, and
the junction itself placed carefully at the center of the flask. The
junctions were brazed in the edge of a Bunsen flame by holding in metal
tweezers immediately back of the point to be brazed. Outside, the tubes
ended in capsules containing the junctions with the copper lead-wires;
these dipped into the water bath containing the bulb. The tubes were
sealed by a mixture of beeswax and resin where they opened into the flask.

The arrangement by which the gas was sent into the flask is shown in
Fig. I. Air was taken from the laboratory compressed air system, passed
through two bottles of concentrated sulphuric acid, then over a con-
siderable' length of solid caustic soda to remove carbon dioxide, and
finally over phosphorus pentoxide. Hydrogen was obtained electro-
lytically and passed through the same system; further purification was
deemed unnecessary, in view of the fact that the question of density is
not involved in the experiment. The pressure gauge, 0, on which the
excess pressure was read, is of tubing 2.5 cm. in diameter, filled with light

K.X.1

SPECIFIC HEATS OF HYDROGEN.

527

transformer oil. With cocks a and c closed, the mercury could be raised
in J2 to just the right height as indicated by the accessory mercury gauge,
M, so that on opening the cock c, the oil moved less than a centimeter.
There is thus practically no change in the gauge reading after the bulb is
filled, due to hanging of oil on the walls. The inverted U-tube through

Fig. 1.

which the expansion takes place serves in the case of hydrogen as a trap
to prevent air from being carried back into the bulb during the surges
incident to the expansion.

The thermal E.M.F. developed by the expansion was measured by a
null method with a Wolff potentiometer. An E.M.F. of the order of
.002 volt was applied from a storage cell which was compared with a
Weston standard before and after each series of observations. A ballistic
galvanometer of the D'Arsonval type was used. It was read by a
telescope and scale at a distance of 3,6 meters; the angle of deflection
was doubled with practically no loss of light by the simple device of
throwing the beam back upon the galvanometer mirror from a small
stationary mirror about 10 cm. in front of it. Under these conditions
the galvanometer sensibility was such that 3.8 X io~* volts corresponded
to I mm. deflection. The constant of the thermoj unction between 0°
and 20° is 3.707 X lO"^ volts per degree; consequendy an equilibrium
temperature could be read directly to .001° and estimated to .0002°.
In order to eliminate spurious motion it was found necessary to mount the
galvanometer on a Julius suspension,* which was built very nearly

■Ann. d. Phys., 56; 151, 1S95.

528

MARGARET CALDERWOOD SHIELDS.

rssooND

according to the original specification, the whole platform weighing
about II kg. ; the situation proved too severe a test of even this admirable
device, so that observations were made only during the quieter portion
of the day.

The method of procedure for a single determination was as follows:
The bulb was filled and the cock b closed at some definite pressure, lo
to 50 cm. of oil in excess of atmospheric pressure. The potentiometer
was kept balanced till it was certain that the gas had attained the tem-
perature of the bath. Then the thermoj unction circuit was opened,
and the potentiometer resistance across which it is shunted changed to
one or two ohms less than that which would presumably balance the
E.M.F. developed by the expansion. The two-way cock, d, was turned
to cut off the pressure gauge; then on one beat of a metronome, d was
turned to open the bulb to the atmosphere, and on the next beat the
potentiometer was closed. An instant backward throw of the galvan-
ometer of I to 5 mm. is observed before it starts forward with rising
temperature. The pressure gauge is read immediately, and the process
is repeated for identically the same pressure with a potentiometer
resistance different by an ohm. From six to ten such observations are
used to fix a line from which the equilibrium resistance may be found by
extrapolation. Data for one such observation are shown in Table I.

Table I.

AT.

>«.

Potttotiomtttttr X.

OalT. Throw.

73.990 cm.

12.60 cm.

23(2

2.8 mm.

18.64*' C.

12.64

4.9

12.59

2.3

12.60

5.0

12.57

2.8

73.945

12.55

4.6

18.68*

Temp, oil 21.7° C.
Ap = 10.88 gm./cm.«

24.21 Q = Ro
7384.3 Q ^ Ro{ Weston cell against storage

Errors Due to Inflow of Heat During Observation. — Both because the
thermoj unction has a finite heat capacity and because the expansion is
oscillatory, it is necessary that the temperature should be measured at
some definite time after the expansion is made. During this interval
there is of course an inflow of heat which holds up the final temperature
of the thermoj unction. The effect of gas conduction and convection
from the walls, which is present even in a 60 liter carboy, as shown by

Jj2"t^l SP^IFIC HEATS OP HYDROGEN. 529

Moody's work, is of course extreme in the i liter bulb; its magnitude,
moreover, will veuy in the present case with the way in which the thermo-
junction 15 mounted in the bulb. There is also an inevitable transfer
by radiation from the walls to the junction, and finally a possible inflow
by metallic conduction. These errors are all proportional to ^T, and
vanish with it. It is consequently necessary that for a single thermo-
junction, with a single interval between expansion and observation of
temperature, sufficient data should be obtained to plot apparent values
of 7 as a function of if, or of the cooling. For expansions as small as
those here employed (none are over 4 per cent.) this must be practically
a linear function. Three different junctions, mounted in different tubes,
were used in air with different metronome rates, and the data so plotted.
Figs, 2 eind 3 show that though the slopes differ widely, the intercepts are

ffcr*

Fig. 2. Fig. 3.

substantially the same. Two series of observations made with the same
junction for ( = .75 sec. and ( = .62 sec. (see Table III., Junction E)
indicate the extent to which the observed values depend upon the time
of observation. Junction D was used with the same time interval in
both air and hydrogen, and a comparison of the D lines in Figs. 2 and 3
shows how much more considerable the effect of heat inflow is in the case
of hydrogen; this is partly due to the shorter time of expansion, and
peirtly to the larger conductivity. It appears then that the data bear
out satisfactorily the assertion that in the value of y obtained by extra-
polation for A^ = o, the effect of inflow of heat by whatever means and

530

MARGARET CALDERWOOD SHIELDS.

Table II.

Air*

AP.

AT.

If Mm

Junction B^ time,

.87 sec

13.98 gm./cm.>

1,014.90 gm./cm.«

292.05*

1.133*'

1.3971

14.00

1,007.50

292.04

1.138

1.3947

•

13.95

1,016.72

291.92

1.135

1.4000

14.21

1,012.46

291.20

1.156

1.3999

14.08

1,012.90

292.77

1.149

1.3985

14.01

1,011.56

292.71

1.151

1.4015

14.00

1,015.15

292.19

1.143

1.4010

14.05

1,014.82

292.18

1.140

1.3973

13.93

1,014.12

292.11

1.138

1.4007

13.91

1,019.70

291.83

1.121

1.3968

1.3988

24.50

1,013.26

292.03

1.970

1.3953

24.51

1,008.00

292.02

1.986

1.3968

24,67

1,012.70

292.80

2.003

1.3990

24.49

1,011.98

292.08

1.967

1.3939

24.47

1,015.26

292.17

1.976

1.3985

24.46

1,014.00

2V2.13

1.969

1.3960

24.45

1,019.90

291.85

1.958

1.3971

24.46

1,018.90

292.01

1.960

1.3964

1.3966

34.95

1,008.42

291.97

2.793

1.3929

35.12

1,012.68

292.79

2.812

1.3949

34.95

1,014.93

292.22

2.787

1.3949

35.00

1,012.13

292.04

2.811

1.3964

35.12

1,011.60

292.80

2.812

1.3943

35.04

1,012.86

292.66

2.805

1.3950

34.93

1,015.40

292.14

2.775

1.3932

35.02

1,019.30

291.96

2.760

1.3913

1.3942

Junction D, time,

.87 sec

10.58

1,023.31

291.16

.857

1.4013

10.73

1,018.60

291.25

.872

1.4008

10.60

1,009.80

290.81

.871

1.4028

10.66

1,016.82

291.12

.869

1.4021

10.79

1,012.50

291.40

.885

1.4027

1.4019

19.72

1,019.70

292.24

1.602

1.4024

19.43

1,010.30

290.79

1.597

1.4007

19.50

1.016.52

291.10

1.580

1.4015

19.66

1,016.15

291.37

1.600

1.4031

19.54

1,023.76

291.34

1.569

1.3998

19.61

1,011.50

291.46

1.596

1.4006

1.4013

28.14

1,021.57

291.16

2.261

1.4023

28.38

1,02U0

292.30

2.268

1.3971

Vot.X.1

SPECIFIC .

HEATS OF HYDROGEN.

531

Table II. — Continued.

LP,

Ar.

Mean

Junction D, time,

.87 sec

28.12 gm./cm.«

1,010.70 gm./cm.«

290.78^

2.284*

1.4032

28.24

1,016.56

291.14

2.277

1.4017

28.39

1.016.30

291.31

2.287

1.4008

1.4010

37.11

1,020.50

292.26

2.981

1.4025

36.88

1.009.35

290.82

2.954

1.3988

37.03

1,016.42

291.26

2.972

1.4017

1.4010

Junction G, time,

1.0 sec

10.79

1,019.06

291.70

.874

1.3987

10.94

1.011.28

291.95

.892

1.3974

10.87

1,006.80

291.85

.894

1.4000

10.83

1,024.61

291.77

.874

1.3992

10.89

1.005.43

291.89

.894

1.3982

10.74

1.012.37

291.72

.878

1.4000

10.88

1.022.30

291.78

.878

1.3980

10.71

1,014.03

291.29

.869

1.3973

10.81

1.003.14

291.40

.888

1.3979

10.88

1,005.59

291.78

.895

1.3991

1.3986

36.91

1.019.06

291.65

2.912

1.3929

37.13

1,010.07

291.90

2.945

1.3908

36.93

1.015.64

291.81

2.928

1.3935

37.13

1.013.47

291.85

2.936

1.3908

37.11

1.010.47

291.82

2.950

1.3921

37.01

1,023.96

291.40

2.890

1.3903

36.87

1,023.41

291.05

2.888

1.3922

36.81

1,017.17

291.90

2.888

1.3905

1.3916

Limi

ting values: B » 1.4019

D = 1.4017

G =

1.4014

Mes

in - i.-^c

however extreme, unless the slope of the line is such as to make its
intercept uncertain, is completely eliminated. Inasmuch as the errors
inherent in the method are thus accounted for, it remains only to examine
the observational errors.

Observational Errors. — ^The error in pu the barometric pressure read
to .005 cm., would be inconsiderable were it not that a single observation
requires at least an hour, and that in that time the barometer often
changes by nearly a millimeter. For the smallest expansion a change of
.5 mm. in pi is equivalent to a change in AT" of .0003**. For this reason
observations were always taken for alternately high and low points on
the resistance-throw line.

Heights on the oil gauge were read to .01 cm. by means of a magnifying

532

MARGARET CALDERWOOD SHIELDS.

Table III.

Hydrogen.

AP.

Ar.

If Mm

Junction C, time,

.65 sec

10.85 gm./cm.«

999.51 gm./cm.«

291.72*

.9or

1.4017

10.52

1,023.78

290.24

.846

1.3997

10.67

1,022.72

291.68

.872

1.4037

10.85

1,022.70

292.90

.887

1.4003

10.67

1,008.67

291.45

.876

1.4006

10.63

1,007.63

291.52

.873

1.4003

10.70

1,016.74

291.26

.873

1.4022

10.74

1,025.44

291.32

.866

1.3999

1.4011

19.60

999.95

291.57

1.612

1.3999

19.60

1,017.70

291.10

1.583

1.4004

19.59

1,011.94

292.86

1.598

1.3994

19.25

1,012.20

292.72

1.589

1.4061

19.55

1,019.30

292.03

1.588

1.4024

19.69

1,021.90

292.92

1.584

1.3970

19.49

1,010.90

291.22

1.589

1.4017

19.61

1,025.44

291.32

1.578

1.4019

1.4011

28.36

1,019.30

292.08

2.273

1.3978

28.27

1,008.33

291.52

2.309

1.4037

28.40

1,018.74

292.47

2.306

1.4043

28.33

1,022.02

292.30

2.276

1.4004

28.41

1,019.03

292.92

2.301

1.4021

28.40

1,014.40

291.97

2.294

1.3999

28.36

1,027.36

291.94

2.267

1.4012

28.16

1,013.80

291.12

2.276

1.4015

1.4012

36.94

1.000.75

291.66

3.003

1.3997

36.88

1,014.46

291.12

2.962

1.4013

37.03

1,015.40

292.54

2.988

1.4018

37.04

1.022.02

292.32

2.977

1.4035

37.14

1.019.03

292.69

2.V86

1.4013

37.14

1.021.90

292.92

2.984

1.4021

36.89

1,014.80

291.09

2.967

1.4020

36.92

1,007.63

291.53

2.982

1.4000

1.4014

Junction D, time,

.83 sec

10.79

1,013.00

291.51

.872

1.3924

10.74

1,000.90

291.12

.876

1.3934

10.62

997.24

290.62

.867

1.3902

10.70

1.008.95

290.92

.865

1.3929

10.72

1,015.20

290.62

.863

1.3963

10.44

1.011.80

290.35

.845

1.3962

10.71

1.006.35

290.77

.869

1.3941

1.3936

Vol X.!
Nas. J

SPECIFIC HEATS OF HYDROGEN.

533

Table III. — Continued.

AP.

AT.

T«

Mean

Junction D, time,

.83 sec

19.61 gm./cm.«

1,003.85 gm./cm.«

291.72**

1.580°

1.3906

19.50

1,005.12

290.62

1.547

1.3847

19.61

1,016.85

290.72

1.549

1.3885

19.56

1,010.20

291.00

1.563

1.3900

19.39

1,010.45

290.40

1.523

1.3824

19.53

1,015.48

291.21

1.545

1.3874

1.3873

28.28

1,003.80

291.72

2.255

1.3871

28.11

1,002.73

290.6J

2.216

13830

28.29

1,016.45

290.70

2.1yO

1.3801

28.17

1,009.00

290.45

2.186

1.3780

28.19

1,006.35

290.77

2.198

1.3789

28.29

1,015.25

291.27

2.194

1.3792

1.3811

37.01

• 1,003.75

291.77

2.869

1.3754

37.07

1,010.45

291.12

2.849

1.3758

36.98

1,015.60

290.62

2.830

1.3766

36.89

1,009.50

290.4?

2.814

1.3725

1.3781

Junction £, time,

.62 sec

10.75

1,021.50

291.60

.877

1.4038

10.76

1,010.68

292.22

.882

1.3993

10.51

1,037.40

290.78

.839

1.4011

10.56

1,036.70

290.42

.841

1.4007

10.64

1,016.90

290.59

.859

1.3977

10.67

986.90

291.23

.898

1.4028

1.4009

19.58

1,021.70

291.68

1.577

1.3997

19.66

1,015.55

291.67

1.607

i.4047

19.29

1,039.70

290.02

1.515

1.3981

19.38

1,018.90

290.81

1.561

1.3998

19.46

1,011.75

291.03

1.575

1.3982

1.4001

28.20

1,022.50

291.72

2.263

1.4010

28.06

1,036.90

290.87

2.216

1.4015

28.05

1,039.75

290.02

2.194

1.3990

28.17

1,015.00

291.16

2.252

1.3960

28.08

1,038.20

290.32

2.200

1.3986

1.3992

•

.75 sec

10.76

1,015.83

291.50

.877

1.4005

10.62

1,017.00

290.92

.861

1.3992

10.69

1,016.50

290.94

.867

1.3995

1.3997

28.11

1,022.60

291.52

2.240

1 .3975

28.04

1,035.40

290.92

2.201

1.3971

28.07

1,039.80

290.09

2.175

1.3938

28.10

1,038.40

290.32

2.190

1.3958

28.07

1.010.23

290.83

2.263

1.3986

1.3965

534

MARGARET CALDERWOOD SHIELDS.

Table III. — Concluded.

AP.

at;

Mmd

Junction G, time,

.65 sec

10.81 gm./cm.*

1,011.58 gm./cm.«

292.22*

.885*

1.3988

10.83

1,013.05

292.12

.885

1.3988

10.76

1,028.84

291.21

.866

1.4008

10.75

1,029.40

291.25

.862

1.3992

10.74

1,031.36

291.41

.859

1.3985

10.64

1,031.24

291.03

.849

1.3979

10.53

1,025.31

290.17

.847

1.4009

10.65

1.029.04

290.41

.850

1.3983

10.85

1,027.92

291.40

.873

1.3999

1.3992

28.20

1,025.75

290.69

2.239

1.3987

28.34

1,028.56

291.17

2.238

1.3>64

28.27

1,029.00

291.42

2.242

13984

28.15

1,030.82

291.36

2.225

1.3977

28.13

1,031.48

290.77

2.208

1.3947

28.02

1,024.90

290.21

2.215

1.3967

28.13

1,025.94

290.12

2.214

1.3950

28.30

1,026.30

291.40

2.236

1.3950

28.13

1,023.41

291.14

2.233

1.3966

Limiting values:

C
D
E
G

1.4012
1.4011
1.4016
1.4011

Mean - J.40J2

glass and a small lamp to illuminate sharply the meniscus from beneath.
A given pressure could be duplicated to about .05 cm. A temperature-
density curve was obtained for the oil by the specific gravity bottle
method which must be accurate to i part in 8,000. It is to be noted,
however, that an error in the density of the oil, as also an error in the
calibration of the thermojunction, these errors being proportional respec-
tively to A/> and AT, do not appear in the final extrapolated value of 7
at all. The maximum error in Ap may fairly be taken as .01 gm./cm.^;
this corresponds to an error in 7 of .0004 for the smallest expansion and
.0001 for the largest.

The water bath, while it contained no thermostat, was large enough
that the temperature was constant to less than .1** during an observation,
the temperature being read to .01® on a mercury thermometer which
had been calibrated against a Baudin standard thermometer. The
uncertainty here introduced in 7 is only .0001.

The major difficulty, of course, lies in the determination of AT. The
galvanometer throws are so rapid, particularly in hydrogen, where the

Na*^*! SPECIFIC HEATS OF HYDROGEN. 535

conductivity is six times greater than in air, and the temperature change
correspondingly rapid, that they cannot be read with precision. In each
throw are involved the questions of the initial balance of the potentiom-
eter, of the duplication of pressure, and the duplication of the time
interval between opening the bulb and closing the potentiometer circuit.
This last element of variation could be eliminated by the introduction
of an automatic key, but in view of the other more considerable items
this seemed unnecessary. The equilibrium resistance is determined from
eight or ten throws, and may moreover be partially corrected if the mean
slope of the line in question is already known from preceding observations.
These resistances are probably obtained to the nearest .i ohm, making
a maximum uncertainty of .0006 and .0025 in 7 for the largest and least
coolings respectively.

The total possible error this accounted for in a single observation is
larger than one would wish it. The actual mean deviation of observa-
tions in one group is, however, scarcely more than in Partington's data,
and with a sufficient number of observations the slope of the y-Ap
line must be obtained with fair precision. It is also to be emphasized
that several errors operate to modify the slope of the line without affecting
the intercept, since for A/>= o, log (^1/^2) =^ o also. The worth of the
work should therefore rather be judged by the variation in the inter-
cepts. The three determinations for air and the five for hydrogen show
a mean deviation in each instance of .0002; this may therefore fairly
be taken to represent the probable error. (See summaries of Tables II.
and III.)

Theoretical Correction. — ^The value of 7 obtained from the ideal gas
equation must be corrected in the case of air for departure from the
ideal gas laws. The original Lummer and Pringsheim method was to
compute the absolute temperature using — 272.4** C. as zero. This is
numerically equivalent to the method used by Partington, who computes
the correction from the Berthelot equation in the form,

y yi Pi — Pt

where yi is computed from equation (2), using ^ = / + 273.09. For
the present purpose the limiting value of the correction term as pilPt » i
is required. This is readily found by substitution of (pi/pt) ^^"^^ '"^ for ^1/^2,
and differentiation to be

a 7 — 1
pii^e 7 *

536 M49C4MET CALDERWOOD SHIELDS.

1^1 I. _ -40170 \

7 yiV ptH /•

The mean values of the critkal constants of air as determined by
CMszeirski and by Wroblewski give a = .356. It follows that for one
atmosphere and 20** the correction factor is .99912 and the corrected
value of 7 is 14029. The correction for hx-drogen is weH beyond the
limit of observation.

Discussion of StsuUs, — The final values obtained from this investiga-
tion are for air 14029, and for h>xlrogen 14012.

Inasmuch as 7 for air is alreadv known with considerable certaintv,
the observ-ations in air are to be regarded as a test of the prccisioo obtain-
able in a smaD flask. A critical summary of the older work appears in
Moody's paper, which points to the conclusion that neither the vekxity
of sound, nor the Oement and Des Ormes method, nor other direct
application of Reech's theorem are capiable of \-ielding as precise results
as the Lummer and Pringsheim method- There are now three deter-
minations by this method, as follows:

L^aacr and Pnacafaciai 1.4025

Moody 1-4005=

PtftiBgtaa ...1.4032

The close concordance between the present ^-alue and these which
obtained in large carbo>3, is a highly satisfactory \'indH:ation of the
applicability' of the method to small flasks. It is interesting to repeat
in passing what others ha\^ called attention to, that, aside from the
internal work, this high \-alue for air is amply accounted for by its I
per cent, argon content. Leduc's formula,* for example, gi^^es 1 4015
for a mixture 99 per cent, of which has 7 = 1 400 and I per cent. 7 = 1.67.

The only available data for h>-dnx:en are meager and conflicting.
Oddly ecough, only three attempts hax-e been made to measure the ratio
of the specific heats of h\-drogen. ManeuxTier,* from direct application
of Reech's theorem. ga\^ for h>-drogen 1.3S4, somewhat less than his
value for air, 1.392, but he frankly states that he had not been able to
secure the same consistency in h>-drogen as in air. The exceedingly
carefiil mork of Lummer and Pringsheim. hoire\Ter, ga\^ 140S4 as com-
pared with 14025 for air. Mercer in the same small flask found 1.398

- M»dys pj.t-jtiei ral-je, I -tail, has be«i irurrraje-d by ibe tbecrKxal correction .ooia.
pctnt^i zzz br Par::-g::e, and ii his be«; iecrease-i by .oc,^c, because the ra.*a?vT« aror
mh,ch be *dic^ had beec airtsady i=c:=i«L szmistakably it a;?p«ftrs to the asthor. in the
al:pe r* his lizic

VouX.
Na

,^•1

SPECIFIC HEATS OF HYDROGEN.

537

for hydrogen and 1.392 for air. The work of Eucken on C» and of
Scheel and Heuse on Cp likewise give values of 7 distinctly higher for
hydrogen than for air (see Table IV.). The weight of existing evidence
is therefore contrary to the present conclusion that 7 for hydrogen is
close to its theoretical value according to the kinetic theory. With the

Table IV.

Observer.

Cp «t 9eP In i^ C«l.

a. Air

Rcgnault'

1.4008

.2408

Lummer and Piingsheim

1.402S

.2400

Swann*

1.3994

.2410

Scheel and Heuse*

1.4013

.2408

Moody

1.4003

.2409

Partington

1.4034

.2396

Shields

1.4029

.2399

b. Hydrogen

Lummer and Pringsheim

1.4084

3.400 (16*)

Scheel and Heuse

1.4075

3.406 (16**)

Shields

1.4012

3.443

> This is after correction by Scheel and Heuse. see Ann. d. Phys. 40: 486.

* Phil. Trans. Roy. Soc., 210: 199. 1909.

* Loc. cit. Scheel and Heuse's and also Swann's values are restated, using / » 4.187.

idea that the discrepancy might be explained if y were a much more
rapidly changing function of the temperature in hydrogen than it is
known to be in air, in which case insufficient care had been exercised in
controlling the temperature of the water bath, a series of observations
were taken with the bulb in an ice bath. This series yielded 1.4006,
however, and there appears no reason to discredit the data of Table III.
on that score.

Compatison with Data on Cp. — ^Alongside these direct determinations
of 7 it is instructive to assemble once more the values of 7 obtainable
from observations on Cp. These latter may be computed in either of
two ways: (a) from the relation Cp — Cv =^ R, corrected in accordance
with a chosen equation of state, as was done by Scheel and Heuse and
by Partington; (&) from the relation.

C„=-;

BapPoapVQ,

Jy-i

employed by Moody, which is as universally rigorous as the thermo-
dynamic theorems out of which alone it is derived, and all factors of
which are known with extreme precision. The second method has been
used in the computations herewith presented, the required constants
being chosen as follows:

538 MARGARET CALDERWOOD SHIELDS. ^SS

For air:
p = 13.595 X 980.616 X 76 (Landolt-Bdrnstein Tables).
^0 = 273.09, chosen by Berthelot as the thermodynamic temperature of

melting ice (Zeit. fiir Elektrochemie, 10 : 621, 1904) and in agreement

with the more recent work of Onnes, Richards, and Witkowski on the

pressure and volume coefficients of hydrogen, and of Travers on helium,
a, = .0036700, Chappuis' value reduced from 1,000 mm. to 760 nmi.

(Trav. et Mem. du Bur. Int. des Poids et Mes., 13: 190, 1903).
ap = .0036713, likewise Chappuis' with the same reduction (loc. cit.).
/ = 4.187 X 10^ ergs per 15® cal. (Ames, Congres Int. d. Phys., i: 178,

1900).
Po «= .00129278, Regnault's value reduced to latitude 45**.

For hydrogen:
a, = .00366256, Chappuis (loc. cit.).
ttp = .0036606, Chappuis, reduced to 760 mm.
Po = 8.9876 X io~*, the mean of Regnault's, Jolly's and Morley's values

as quoted by Berthelot (loc. cit.).
The table gives observed values in Clarendon type, computed values in
ordinary type.

In the case of air the computed values of Cp are, with the exception of
Moody's, lower than the observed. This may be attributed to uncer-
tainties in the data which are the basis of the computation; it is to be
noted, however, that Partington's method yields values slightly lower
yet. For hydrogen the two methods are identical, as would be expected
when there is no question of equation of state.

II. At - 191^ C.

The same one-liter flask was placed in a liquid air bath — an open
metal can well packed in cotton so that two liters of air sufficed for one
and a half hours' work — ^and it was found that observations could be
obtained with surprising ease and precision. The fixed junction was
kept in ice, and in order to balance the large electromotive force due to
190°, a second potentiometer was applied to the thermo junction circuit
having a fall of 1.5 volts through about 800 ohms; 3 ohms were sufficient
almost to balance the thermal E.M.F., and the close adjustment was
made on the Wolff potentiometer arranged as before. In this way the
resistance in the thermoj unction circuit was kept small, and the ther-
mometric arrangement was quite as sensitive as at 20®. It was more
difficult to reproduce pressures, and the temperature of the bath inevit-
ably changed by about .5® between beginning and end of a complete
observation. The procedure was in other respects the same as at 20**.

VOL.X.

Na s.

]

SPECIFIC HEATS OF HYDROGEN.

539

The tempera ture-E.M.F. line was obtained very simply by measuring
the E.M.F. in pure oxygen (used air was accumulated until a liquid was
secured which maintained its temperature to 1/4000 for seven hours) and
then in new air, the temperature of the latter being obtained from Baly's
data^ by displacing the line somewhat to accord with Henning's tempera-
ture for oxygen* (— 183.00**), and Fischer's for nitrogen^ (— 195.67**) C.
This method was checked closely by calibrating a platinum thermometer,
and then calibrating the thermo junction against it. The calibration
constant of the junction in this region is 1.68 X lO"* volts per degree.

The data obtained are shown in Table V.

Table V.

AP.

AT.

Mean y.

13.84 gm./cm.'

1.011.26 gm./cm.*

81.05* A.

.394''

1.559

14.01

1.020.51

80.35

.394

1.564

14.39

1.018.89

81.85

.411

1.561

14.61

1.013.50

83.33

.427

1.560

14.20

1.006.70

82.52

.419

1.571

1.563

36.79

1,020.80

81.50

.968

1.509

36.82

1.018.89

82.43

.978

1.506

37.11

1.006.35

82.30

1.015

1.521

36.56

1.007.05

82.33

1.011

1.530

36.43

1,012.90

82.51

1.011

1.535

1.520

7 for P i2. o, 1.590

The correction factor for hydrogen at 82** A. is i.ooii. Hence the
final value in the ideal gas state is 1.592 at a mean temperature of 82**,
as compared with Scheel and Heuse's value 1.595 at 92**, and Eucken's
1.605 at 92**, or 1.624 at 82°. (These latter are computed from the rela-
tion, Cp — C9 ^ R, corrected in accordance with Berthelot's equation,
as indicated by Scheel and Heuse.) The value here obtained might be
slightly smaller except for the unfortunate circumstance that the mean
temperature of the bath during observations at P = 14 gm./cm.* is
lower than at 37 gm./cm.*. The smaller value of 7 agrees more closely
than Eucken's, however, with the Planck-Einstein formula for C». With
a specially designed thermostat, this method might well be made a
more precise method of investigating specific heats of gases at low
temperatures than direct measurement of either Cp or C».

At the conclusion of this work the author wishes to testify to her

» Phil. Mag. (5). 49: 5x7-
* Ann. der Phy. (4), 35: 761.
> Ann. der Phy. (4), 9: 1149.

540 MARGARET CALDERWOOD SHIELDS. ^S»

appreciation of the friendly interest of all the Ryerson staff during its
progress, and to her especial indebtedness to Professor Millikan for the
constant encouragement and helpfulness of his oversight.

Summary.

1. Observations by the Lummer and Pringsheim method in a one liter
flask have been found to yield for air, y = 1.4029, a value in close agree-
ment with already accepted values.

2. For hydrogen, 7 is found to be 1.4012 at 18° C, 5. per cent, lower
than previous values, with no apparent explanation of the divergence.

3. For hydrogen at — 191° C, 7 becomes 1.592, in general accordance
with the quantum theory of specific heats.

Rybrson Physical Laboratory,
University op Chicago,
May, 191 7.

No's^] NOTES ON MELDE'S EXPERIMENT. 54 1

NOTES ON MELDE'S EXPERIMENT.

By Arthur Tabbr Jonbs and Marion Eveline Phsu>s.

I. Vibration Form of Elements of the String.

Introduction. — Melde* studied the vibrations of a string of which one
end was fixed and the other attached to one prong of a tuning fork.
When the string was perpendicular to the axis of the fork and in the
plane of its prongs he spoke of the system as being in the "parallel"
position, and when the string was perpendicular to the plane of the
prongs he spoke of it as being in the ** transverse " position. In the cases
of the simplest motions the frequency of the vibration which the fork
imposes upon the string is equal to that of the fork for the transverse
position and half that of the fork for the parallel position.

We have found that by using for the string a black fish cord in which
white dots had been woven it is possible to see very clearly the paths of
the separate elements of the string and even to obtain photographs of
these paths.

The Parallel Position. — In this case if the end of the string which is not
attached to the fork is fixed in position, the vibration of the fork produces
a periodic change in the distance between the ends of the string. Thus
the motion of each element of the string is compounded of a longitudinal
component which has a frequency equal to that of the fork, and a trans-
verse component which has a frequency twice that of the fork. The
velocity with which a change of tension is propagated along the string
is usually so great compared with the velocity with which transverse
waves are propagated that we may regard the tension at any instant as
being the same at all points of the string. In a system in which the
damping is small the string passes through its equilibrium position when
the tension of the string is greatest or least* — usually when it is greatest —
so that the paths of the elements of the string are usually parts of para-
bolas with their convex sides turned toward the fork. Fig. 2 shows these
parabolic arcs.. The fork is at the right. It will be observed that the
longitudinal component of the motion is practically constant from one
end of the photograph to the other.

> Pogg. Ann., 109, p. 192, 1859; III* P> 513* i860.
* Rayleigh, Theory of Sound, ed. 2, Vol. i. p. 84.

542 ARTHUR TABER JONES AND MARION EVEUNE PHELPS.

fSBOOHB

LSbib.

Raman^ has shown that the string may be maintained in vibration

with a frequency m/2 times that of the change in tension, where m is any

int^er. For w = i we have the case studied by Melde. Raman gives

pictures of a string which is vibrating with two of these frequencies

simultaneously, one of the two being alwaj^ the frequency for which

m = I. The striking fact about Raman's cases is that the string vibrates

in one loop for both frequencies.

Fig. 3 shows a case in which the frequency of the string equals that

of the fork. This frequency ratio is shown by the fact that the paths

are nearly straight lines. Thus these lines make immediately evident

what Raman has deduced* with the aid of his Fig. 7.

Fig. 4 shows a case in which the string is vibrating simultaneously

with two loops and with five, but in which the frequencies are not in the

ratio of two to five. A node for the form with two loops is near the

middle of the photograph, and nodes for the

form with five loops are a little beyond the

ends of the photograph. The frequency of

the vibration with two loops is half that of

the fork, as is seen from the fact that near

the ends of the photograph the curves ap*

proach arcs of parabolas. The frequency of

the vibration with five loops is three halves

that of the fork, as is seen from the fact that

the curves in the photograph have in general

the forms shown in Fig. i. a is the form

toward the left of the photograph, b near the

middle, and c toward the right. These curves are drawn from the

equations

X = — 1.25 cos 2ptf

y ^ 3 cos />/ + 3 cos ipi,

y = 3 cos ipt,

y = — 4 cos pi + 2 cos 3/>/.

b
Fig. 1.

for all curves
for a
for 6
for c

For Fig. 4 the string was about 70 cm. long, of linear density about
2.2 mg./cm., and was driven by a fork of frequency 100 vd. The tension
could be varied by turning a screw eye, around which the fixed end of
the string was wrapped. When the photograph was taken the string
was vibrating in a plane, but it is quite as easy to obtain this figure
when the string shows two loops in one plane and five in a perpendicular
plane.

» Phys. Rev., 35. p. 453, 1912.

Physical Review, Vol. X.. Second Series. Plats I.

November. 1917. To face page 541.

A. T. JONES AND M. E. PHELPS.

No's^*] NOTES ON MELDE'S EXPERIMENT, 543

We have here then a case similar to those studied by Raman, in that
the string is simultaneously executing vibrations with two frequencies
which are not proportional to their respective numbers of loops. This
seems to mean that two transverse disturbances are simultaneously propa-
gated along one string tuith different velocities.

The Transverse Position. — In this position the frequency of the string
is the same as that of the fork. For certain tensions the paths of the
elements of the string may become circles, in planes perpendicular to
the length of the string. Figs. 5 and 6 show the string when vibrating
in this manner. For Fig. 5 the camera was placed at an oblique angle
to the string, so that only the part near the middle of the photograph
was well focused. For Fig. 6 the camera was perpendicular to the string.
At the extreme ends of Fig. 6 the circles begin to show as very narrow
ellipses.

The explanation of these circles appears to involve a force of double
frequency. Consider the system in the transverse position and the
string vibrating in a plane. When the string is in either of its extreme
positions its length — ^and therefore its tension — ^is greater than when
it passes through its equilibrium position. Thus the tension of the string
is subject to a periodic change of which the frequency is twice that of
the vibration of the string. This periodic change of tension can maintain
a transverse vibration of the string with a frequency* which is the same
as that with which the string is already vibrating. For brevity we will
call the vibration which is immediately due to the motion of the fork the
** motional" vibration, and that which is due to the change in tension
the "tensional" vibration.

When the damping is small the tensional vibration will pass through
its equilibrium position when the tension is greatest or least' — usually
when it is greatest. Thus the tensional vibration will either be in phase
with the motional vibration or will differ from it in phase by 90*^ —
usually the latter. If the phases are the same, or if the planes of vibra-
tion happen to be the same, the two motions compound into a plane
motion. But if the phases differ by 90*^, and the vibrations happen to
lie in perpendicular planes and to have equal amplitudes, the resultant
motion will be circular.

Now for the motional vibration the end of the string which is attached
to the fork does not lie quite at a node, whereas for the tensional vibration
it does. Thus the distance between nodes is slightly different for the
two motions. The result is that near the nodes the circles pass over into

1 Rayleigh, Theory of Sound, ed. 2, $ 686.

* Rayleigh, Theory of Sound, ed. 2, Vol. i. p. 84.

544 ARTHUR TABER JONES AND MARION EVELINE PHELPS. [^JS

ellipses — ^with major axes approximately parallel to the motion of the
prong on one side of the node and perpendicular to it on the other. These
ellipses are easily observed.

It may be thought that the change of tension produced by the change
in length of the string during vibration would not be sufficient to main-
tain a vibration of sufficient amplitude to produce the above effect.
But the length of one wave of a string which has the form of a sine curve
of wave-length 50 cm. and amplitude i cm. is about 2 mm. greater than
its horizontal projection, so that the increase in tension would in this
case be the same as that due to an amplitude of a millimeter in the
prong of a fork in the parallel position. It is also to be remembered that
the force which produces the motional vibration is applied farther from
a node than that which causes the tensional vibration.

II. Rotation of the Pulley.

J. S. Stokes* observed that when the Melde apparatus is in the parallel
position and the string passes over a pulley, the pulley may sometimes
be set into a more or less steady rotation. Raman and Apparao* have
also observed a rotation of the pulley. When the string passes hori-
zontally from the fork to the pulley and then downward we will call a
rotation in which the top of the pulley moves toward the fork a rotation
toward the fork, and a rotation in which the top of the pulley moves
away from the fork a rotation away from the fork. Stokes found that
when the pulley was rotating toward the fork, waxing the string near
the pulley led to rotation away from the fork, and cutting off the wax
with oil restored the rotation toward the fork.

Shortly afterward one of us found that if the string passed through a
small hole placed at the node nearest to the pulley, a rotation toward
the fork could sometimes be produced by moving the hole a short distance
toward the pulley, and a rotation away from the fork by moving the
hole a short distance away from the pulley. Thus the pulley turned
away from the fork when the segment which ended at the pulley was
longer than it would be for the free vibration of the string, and toward
it when the segment was shorter. We have checked this result by
calculating the lengths of the loops of the free vibration for different
loads, and then applying several loads which were in the neighborhood
of those calculated. For the smaller of these loads the rotation is in
general away from the fork and for the larger toward it.

It had seemed possible that the change in the sense of rotation observed

» Phys. Rev., 30, p. 659, 1910.
• Phys. Rev., 32, p. 307. 191 1.

No!"^^] NOTES ON M ELBE'S EXPERIMENT. 545

by Stokes might be due to a stifTening of the string by the wax and a
consequent change in the distance between the nodes of the free vibra-
tion, but if that were the explanation his rotations would apparently
not have been in the senses he observed.

For short strings it is difficult, if not impossible, to obtain rotation
away from the fork, whereas rotation toward the fork is usually easy to
obtain. Sometimes the rotation is very slow and sometimes the pulley
makes several turns in a second. Frequently the sense of rotation may
be changed by simply changing the amplitude with which the fork is
vibrating, a small amplitude giving rotation away from the fork and a
large amplitude rotation toward it. Rotation away from the fork
appears to occur only when any motion that the string may have is
pretty steady, whereas much irregularity in the vibration of the string
is almost sure to cause an irregular turning of the pulley toward the fork.
Moreover a rotation of the pulley in either sense may be obtained when
the string is not, at least visibly, executing any transverse vibration at all.

When the string vibrates transversely this transverse motion may
in certain cases be just sufficient to take up the slack given by the
approach of the prong, so that no vibration or change of tension is
transmitted to the pulley or beyond it. In a case which appeared to be
of this sort we have measured roughly the amplitude of the motion of the
string, assumed the string to be displaced in a sine curve, and have
found that the length of this sine curve really did exceed its horizontal
projection by an amount which checked, within the limits of experimental
error, with the double amplitude of the tuning fork — in this case 7 mm.

The rotation of the pulley is doubtless due to an intermittent slipping
of the string along the pulley at a phase of the vibration at which the
friction between them is too small to supply the acceleration which
would be necessary to prevent the slipping. A beginning has been made
at the theory of the rotation, and that together with further experimental
work will form the subject of another paper.

Smith Collbgb,
June 19, 1917*

546 H. J. VAN DER BIJL,

THEORETICAL CONSIDERATIONS CONCERNING IONIZA-
TION AND SINGLE-LINED SPECTRA.

By H. J. VAN DER BiJL.

IN the following an attempt is made to give an explanation of some of
the rather conflicting results on the ionization and characteristic
radiation produced by the passage of electrons through gases and vapors.
Most of the investigations of these phenomena were performed with
mercury vapor, but the following considerations will in the main apply
also to other monatomic gases and vapors.

The most important result of the experiments of Franck and Hertz^
is that collisions of electrons with molecules of mercury vapor are elastic
until the electrons have acquired energy equivalent to 4.9 volts. After
having dropped through this voltage the electrons lose all their energy
on collision and at the same time energy is radiated. This radiated
energy Franck and Hertz identified as the single-line 2536 A.U. This
shows that the mercury atom does not take any energy from the colliding
electron unless the latter has a definite minimum amount of energy to
give to the atom. Furthermore, in view of the fact that, according to
the Planck-Einstein relation Ve = hv, the frequency of the line 2536
corresponds to 4.9 volts, the experiments of Franck and Hertz gave
evidence in favor of the quantum theory. They concluded from their
experiments that when the electrons have dropped through 4.9 volts,
ionization of the mercury vapor sets in. By using LenardV method of
picking out the positive ions, they actually observed what api>eared to
be ionization at 4.9 volts. This result was later confirmed by McLennan
and Henderson,* Goucher* and others. McLennan and Henderson also
found that the single line 2536 was emitted when the atoms of mercury
vapor were bombarded by 4.9 volt electrons, and established a similar
result for cadmium and zinc.

These results presented two difficulties: Firstly, although the result
that the collision of an electron with an atom of a monatomic gas is
elastic when the electron collides with energy less than a certain definite

> Verh. d. D. Phys. Ges., i6, 457 and 512, 1914.

* Ann. d. Phys., 8, 149, 1902.

* Proc. Roy. Soc.. A. 91, 485, 191 5.
■• Phys. Rev., 8, 561. 1916.

No's^'] IONIZATION AND SINGLE-LINED SPECTRA. 547

amount is in conformity with the quantum theory, such conformity
does not exist if ionization as well as radiation is produced by the trans-
ference of this definite amount of energy from the colliding electron to
the atom. Secondly, the quantum theory requires that the transference
of this amount of energy should give rise to the stimulation of a single
line. But if, on the other hand, a stream of colliding electrons is used
(as is always done) the emission of a single line is not compatible with
Bohr's theory of the atom. It is because of these definite and important
questions that experiments along these lines may reasonably be expected
to furnish valuable evidence regarding the validity of the quantum
hypothesis and particularly Bohr's theory of atomic structure.

As regards the first question, viz., the production of ionization simul-
taneously with the stimulation of the 2536 line in mercury, I pointed out
that the ionization effect observed at 4.9 volts may, under certain condi-
tions, not be impact ionization, but a photoelectric effect^ The stimula-
tion of the line 2536 establishes a source of ultraviolet light in the
discharge tube and so causes a dislodgment of electrons with the attending
phenomena of ionization. On this view, what we might call the ioniza-
tion voltage of mercury vapor would be 10.4 instead of 4.9 volts. This
view has since been confirmed experimentally by Goucher.^

As regards the second question, the reality of the single-lined spectrum
does not seem to have been established. McLennan^ found that the
many-lined spectrum was not produced until the colliding electrons have
acquired an amount of energy equivalent to about 10 volts, this voltage,
according to the quantum relation, corresponding to the line 1188 A.U.,
which is the limiting line of the series of which 2536 is the first member.
This result they also found with the vapors of cadmium, zinc and
magnesium. On the other hand, Hebb and Millikan* find that the mer-
cury arc, emitting its many-lined spectrum, can be made to strike with
any voltage greater than 4.9 volts. I have been informed that at the
April meeting of the Physical Society McLennan reported that by using
dense electron streams as suggested to him by Millikan he had confirmed
the latter's results.

In discussing this matter at the New York meeting of the Physical
Society last December I pointed out that the apparent discrepancy could
be explained away on the basis of the quantum hypothesis of atomic
radiation by considering three factors:

^ Proc. Am. Phys. Soc., Chicago meeting, Dec. i, 1916. I wish to point out here that
since the publication of this suggestion I found that Bohr had himself suggested the possi-
bility of a photoelectric effect to explain the apparent discrepancy between the observed
ionization voltage and that calculated from his theory of atomic structure.

« Read at February meeting of Am. Phys. Soc., 1917. Phys. Rby., 10. loi, 1917.

« Proc. Roy. Soc., A, 92, 305, 1916. * Phys. Rbv.. 9, 371 and 378, 1917.

548 H, J. VAN DER BIJL.

1. As soon as the gas or vapor is brought into a state of excitation the
size of the atoms increases, with a resulting transformation of the atomic
system into one of higher potential energy.

2. When the colliding electrons have acquired sufficient energy to dis-
place electrons from the outermost orbit, the radiation, resulting when
the displaced electrons drop back to any configuration corresponding
to one of lower potential energy than that which the system has acquired
by virtue of the collision, stimulates a photoelectric effect with the result-
ing production of dislodged electrons.

3. The apparent distribution of velocities of the colliding electrons
may, under certain circumstances, produce an appreciable influence.

Let us consider these influences in succession and see in how far they
are capable of lending an explanation of the observed phenomena.

I. That the atomic diameter must increase with the excitation of the
gas or vapor is in strict accordance with the quantum hypothesis, and
in fact, follows as a natural consequence of it. This is easily seen by
considering the simplest case, namely that of hydrogen. This atom has
only one electron and one positive nucleus, the electron being in the
orbit corresponding to the minimum potential energy of the system. We
shall call this orbit i. Now the Balmer series is stimulated by the dis-
placement from (and consequent dropping back to) the orbit 2; the
Paschen series by the displacement from the orbit 3. Only the Lyman
series is stimulated by a displacement from the first orbit. Thus, if the
electron is displaced from 2 to 3 and drops back to 2 we obtain the first
line of the Balmer series; 2 to 4 gives the second line of the Balmer series
and so on. It is therefore evident that in the state of excitation of the gas
there must be many orbits outside the first which contain electrons,
some atoms having, at any particular moment, electrons in the first
orbit, others having their electrons in the second, others in the third orbit,
and so on.

It is easily seen what is the cause of this increase in the potential
energy of the whole system. If there are only a few colliding electrons
the chance of this happening would be very small. But if a dense stream
of electrons is used, then a bound electron which has been displaced
by a colliding electron to an orbit corresponding to higher potential
energy stands a good chance of being knocked out again by another
colliding electron before it has had a chance to drop back to its original
orbit and so will emit a line which belongs to an entirely different series
from that to which belongs the line it would have emitted if it had had
a chance to return to its original orbit. Thus, if the electron of a hydro-
gen atom is displaced from orbit i to orbit 3, it would, if it could return

No's'!^'] IONIZATION AND SINGLE-LINED SPECTRA. 549

to I, emit the second line of the Lyman series; if, however, it is displaced
again by another electron before getting a chance to drop further back
than orbit 2, it would emit the first line of the Balmer series.

The same holds true for mercury. It follows from the experiments of
Franck and Hertz that the most loosely bound electrons in the mercury
atom in the normal state of its vapor are those which require a minimum
amount of energy equivalent to 4.9 volts (2536 A.U.) to displace them
from their position of equilibrium. In the light of the Bohr theory
this would mean the outermost stable orbit which contains electrons in
the normal state of the vapor is that which corresponds to a potential
energy equivalent to 4.9 volts less than the next succeeding orbit
(reckoned from the center of the atom outwards) and which requires
10.4 volts to completely detach an electron from it. Now, as a matter
of fact, the mercury spectrum shows many lines of much greater wave-
length than this ultra-violet line 2536. These lines must be stimulated
by displacements through orbits of greater potential energy than that
corresponding to 2536. In other words, a smaller amount of energy
than 4.9 volts is required to stimulate them, although the experiments
of Franck and Hertz show that the smallest amount of energy that can
cause any stimulation at all is equivalent to 4.9 volts. This all means
that if the frequency of collisions of the impacting electrons with the atom
is small, the line 2536 will be radiated when the colliding electrons have
dropped through 4.9 volts. But if a dense stream of electrons is used
some of the bound electrons that have been displaced from the outermost
orbit which in the normal state of the vapor contains electrons (say
orbit n) to the next succeeding orbit (n + i), will be displaced again by
other colliding electrons before getting a chance to drop back to their
original orbit n, and some of these may be displaced from n + 2 before
dropping back to n + i, and so on. Hence, if we got an instantaneous
picture of the vapor when bombarded by a dense stream of electrons,
we would see atoms of various sizes, some being several timeslarger than
the atoms in the normal state of the vapor.

Now, the energy necessary to displace an electron from the orbit
n -t- I to n -t- 2 is much less than that necessary to cause a displacement
from n to n -f I, the energy decreasing with the number of the orbit.
This follows from Kossel's^ frequency relations:

• •• ••• ••• ••• ••• •••

• •• ••• ••« ••« ••• «••

* Verh. d. D. Phys. Ges.* 16, 953, 1914.

550 H, /. VAN DER BIJL.

where a, /3, etc., represent the number of the lines in the several series
and n and n + i, etc., the difTerent series corresponding to the successive
stable orbits n, n + i, etc. For the X-series of characteristic X radia-
tions, according to Kossel, n = i, the series thus resulting from dis-
placements from the innermost orbit; for the L-series, n = 2, etc. It is
interesting to note that a displacement even from the outermost orbit
which in the normal state contains electrons, does not, for the elements
investigated, give rise to visible radiations; the lowest frequencies that
the normal mercury vapor atom, for example, can give are the ultra-
violet series 2536- • 'iiSS. According to these views the Lyman series
of hydrogen are nothing else than Barkla's X-series of characteristic
X-radiations for hydrogen ; the Balmer series the i-series of Barkla.^

The above frequency relations are general and have been tested by
Kossel for X-radiations and recently by Millikan* for ultra-violet radia-
tion from mercury vapor. Kossel deduced these relations from general
considerations of the manner in which energy transformations are sup-
posed to take place in the Bohr atom. But they also follow directly
from Bohr's equation:

'-{h-hi

of which a number of spectral series have been found to be special cases.
According to this equation v^ is given by

^ \n* (n + i)*/

Similarly,

''^•"^\ii^"(n + 2)V'

"•^^ " ^\(n-f i)»"(n-f2)V'
Hence,

which is Kossel's relation.

From the above consideration of the increase in atomic diameter it
follows that one would not expect to obtain a single-line spectrum unless
the stream of colliding electrons is very attenuated. In such case the

> Other strong evidence for this conclusion is presented in Millikan's recent presidential
address to the American Physical Society.
« Loc. cit., p. 378.

No's'!^'] IONIZATION AND SINGLE-UNED SPECTRA. 55 1

lines of longer wave-length will be so weak that they will not show on the
photographic plate. As the density of the stream of electrons is increased
the intensity of the long wave-length lines will increase. If the electron
stream is very dense, as in the case of the mercury arc, the long wave-
length lines will become very intense. Furthermore, on account of the
increase in potential energy when the atom ** swells" the energy required
to completely detach an electron from the atom becomes less than 10.4
volts. In fact, from Kossel's relation it is seen that the necessary amount
of energy for this may be as low as 4.9 volts and even less. And hence
ionization by successive impacts may take place even at these low vol-
tages. In general the ionization at these low voltages may not be great,
but in the case of a dense stream of electrons as in the mercury arc, it
may be considerable.

2. The second point to be considered is the photoelectric effect due to
the light radiated from the atoms in their attempt to regain their original
configuration after having been disturbed by the impacting electrons.
In the case of devices like Lenard's, in which a third electrode in the
form of a screen is interposed between cathode and plate, most of the
effective excitation, when the applied voltage is about 5 volts, takes
place between the screen and the plate. It is therefore to be expected
that a great part of the photoelectric effect would act on the plate. The
positive ions formed by collision ionization would not have an appreciable
influence on the distribution of potential between the cathode and the
screen, because they are drawn to the plate, which, it will be remembered,
is always maintained at a negative potential with respect to the cathode.
In the case of a two-electrode device the photoelectric effect on the plate,
which is now anode, will not have any influence, since the photoelectrons
cannot come out of the positive plate, or will at least be returned to the
plate when they do come out. The photoelectric effect can, however,
also manifest itself by its action on the neighboring atoms and, as we
shall see below, also on the cathode. Millikan has recently applied the
photoelectric effect produced by the line 2536 on the neighboring atoms
to explain the appearance of the lines of longer wave-length than 2536
on his photographic plates, on the basis that the observed photoelectric
long wave-length limit of mercury is 2800 A.U. He has pointed out,
however, that so far as his argument is concerned it is immaterial whether
the photoelectrically liberated electrons come from the mercury vapor,
from condensed films of mercury or from the substance of the cathode,
and he accordingly leaves the question of their origin entirely open,
insisting only on their being produced photoelectrically. With respect
to this question, it must be remembered that the photoelectric effect

552

H. J. VAN DER BIJL.

rSsooiiD
LSbkxbs.

will exert an influence in producing positive ions from the atoms at low
voltages only in virtue of the swelling of the atoms when the vapor is
brought into a state of excitation. As stated above, it follows directly
from the experiments of Franck and Hertz that the normal mercury
vapor atom does not generally contain electrons in orbits greater than
that corresponding to the series 2536* • 'iiSS, and hence, if there were
no swelling, that is, no increase in potential energy by successive impacts
by electrons or successive stimulation by light, the least amount of
energy that could stimulate any radiation from that atom would be that
which is equivalent to 4.9 volts. And therefore the photoelectric effect
due to the stimulation of the line 2536 by an atom could only cause the
absorption and re^mission of this line by another atom, and would not
assist in the direct multiplication of positive ions.

The emission of electrons from atoms of the vapor cannot be deter-
mined by the photoelectric long wave-length limit of the substance itself.
This quantity is a different thing from the long wave-length limit of the
photoelectric effect on the atoms in the gaseous state of the substance.
In the latter case this quantity is determined by the energy necessary
to completely detach an electron from the atom, whereas in the former
case the energy necessary to detach an electron from the atom must be
very small because of the frequent collisions of the atoms in the solid
state of the substance and the effect of the electrons in the neighboring
atoms, and therefore the photoelectric long wave-length limit of the
solid or liquid is mainly determined by the work which an electron must
do in order to escape from the surface of the substance. The following
table shows the difference between these two quantities for a few sub-
stances. The long wave-length limits of the solids and of the atoms in
the gaseous state of the substance respectively are denoted by Xo and Xc,
and the equivalent voltages by Vq and Ve. The values of Vo are taken
from photoelectric and thermionic measurements, while the values of Ve
are calculated from the convergence wave-lengths Xc of the principal
series, and are assumed, for reasons developed in this paper, to represent
the ionization voltages of the substances.

Substance.

Ko.

^e.

n-^o.

Mercury ....

Zinc

Magnesium .
Calcium ....

2800 (Millikan)
3570 (Richardson)
3750 (Richardson)
3660

1188 (Paschen)
1320 (Paschen)
1336 (Paschen)
1246 (Lyman)

4.44
3.48
3.32
3.4 (W. Wilson)

10.4
9.24
9.13
9.96

6.0
5.8
5.8
6.6

The data available at present are far too meager to warrant any im-
portance being attached to the fact that the difference between Vo and

No's^] IONIZATION AND SINGLE-LINED SPECTRA, 553

Ve for the substances given here is nearly constant. If, however, this
were to be found to be generally true, it would mean that if the ionization
voltage of a substance is less than about 6 volts, or if the substance has
a convergence wave-length of the principal series greater than about
2000 A.U., it should be photoelectrically active in the dark.

The values given in the table show, at any rate, that there is a con-
siderable difference between the energy necessary to detach an electron
from the solid or liquid and that required to detach an electron from an
atom in the gaseous state of the substance. Since, therefore, the long
wave-length limit of the mercury atom in the gaseous state is 1188 A.U.
the line 2536 cannot detach electrons from the normal mercury atom.
All the atom can do is to absorb the light and may reemit it. But in
absorbing it the potential energy of the electron in the atom is increased,
and if it is again exposed to 2536 radiation before getting a chance to
reemit the absorbed light its potential energy will be further increased,
and by a third stage of the process the electron will be knocked out of
reach of the attracting forces of the atom and will be carried away by the
applied electric field. Also, as explained above, the transformation of
the atom into a configuration of higher potential energy will give rise
to the emission of light of different wave-length from the line 2536.

The great difference between the photoelectric long wave-length limit
of a substance and that of the atoms of its vapor carry weight in the
explanation of the maintenance of an arc, say between mercury and iron
electrodes. It means that since a great deal of the light emitted by the
stimulated atoms is of shorter wave-length, and therefore of greater
energy, than the minimum amount of energy necessary to liberate
electrons from the surface of the substance, there must be a copious
emission of electrons from the cathode under the influence of the light
radiated by the atoms, and these electrons must on account of the dif-
ference between Vo and Ve be emitted with appreciable velocity. Thus,
if they are liberated by the convergence line 11 88 they will start with an
initial velocity of about 6 volts; those that are liberated by the intense
line 2536 will have an initial velocity of about 0.4 volt.

3. This brings out the importance of the initial velocities, even in the
case in which a hot filament is used as cathode, and where the arc is
seldom very intense. There is no reason why an appreciable number of
electrons should not be emitted by the hot cathode photoelectrically
once the gas or vapor is stimulated. If, for example, a calcium-coated
platinum cathode is used the line 2536 would liberate electrons from it
with an initial velocity of 1.5 volts. Adding these electrons to those
that are emitted thermionically with Maxwellian velocities, we see that

554 ^' ^' ^^^ ^^^ BijL.

the initial velocities can under certain circumstances have quite an
appreciable influence in maintaining an arc at applied voltages less than
that necessary to ionize the atom, and in fact less than that necessary
to cause any stimulation at all of the normal atom.

It is possible that the comparatively high velocities with which
electrons may be emitted from the cathode under the influence of the
light from the stimulated atoms may account for the discrepancy in
the experimental results on the ionization of helium. Franck and
Hertz,^ Pawlow* and Bazzoni* find the ionization voltage of helium to
be about 20 volts, while Bohr's theory requires that it should be 29
volts. Bazzoni took special care to purify his helium and used a device
which consisted only of a hot-wire cathode of tungsten and a cylindrical
anode. His current-voltage curves show a very sharp increase at about
20 volts, thus indicating the occurrence of impact ionization. While it
is true that there could not have been any photoelectric liberation of
electrons from the anode in his device, it is quite possible that there
might have been such electron liberation from the tungsten cathode.
Remembering that the photoelectric long wave-length limit of tungsten,
according to thermionic measurements of Langmuir, is equivalent to
4.5 volts, it is seen that 20-volt light, which has a wave-length of only
620 A.U., should be capable of liberating electrons from tungsten with
the high initial velocity of about 15 volts, so that when the applied voltage
is 20 the energy of these electrons, on reaching the anode, would corre-
spond to about 35 volts. The current would therefore not only be in-
creased by the extra electrons liberated photoelectrically from the
cathode, but also by the electrons dislodged from the helium atoms by
these high velocity electrons. Adding to this effect the increase in
potential energy by successive impacts, as explained above, we see that
quite a considerable amount of ionization can take place uqder an applied
voltage which is too low to ionize the normal helium atom.

The fact that ionization does not set in until the applied voltage is
20 volts gives, when we consider the effects that manifest themselves
here, a rather striking confirmation of Bohr's theory, because it follows
from his theory that this is just the voltage necessary to displace an
electron from the orbit, which in the normal state contains the two
electrons, to the next succeeding orbit. According to the theory of Bohr
the energy radiated in the formation of single orbit atoms is given by

> Verh. d. D. Phys. Ges., p. 34, 1914.

* Proc. Roy. Soc., 90, 398, 19 14.

• Phil. Mag., 32, 566, 1916.

It&'s^'] IONIZATION AND SINGLE-LINED SPECTRA. 555

where Wq is the ionization energy of the hydrogen atom and F is given by

F = N — 2^ cosec T ,

N being the number of nuclear charges and n the number of electrons
in the orbit. Since for helium N = 2 and n = 2, this gives for the forma-
tion of the helium atom an energy dissipation equivalent to 6.12 Wo
and for the binding of only one electron with the double nucleus, 4 Wo.
Hence the ionization energy of helium is (6.12 — 4)Wo = 2.12 Wo and
the energy necessary to displace an electron to the next stable orbit

2.12PF0I I — - j .

Since Wo is 2.16 X I0"~", this gives 21 volts. It is therefore to be
expected that ionization in helium should start at about 20 volts, because
this is the minimum voltage necessary to cause the swelling of the atoms
and the liberation of the photoelectrons.

It is seen, therefore, that a consideration of the three factors: the
increase in atomic. potential energy by successive impacts, the photo-
electric effect of the light emitted by the stimulated atoms and the initial
velocities of the electrons emitted from the cathode affords an explana-
tion of the results obtained by workers in this field. In particular, the
recently published results of experiments of Millikan and Hebb are just
what is to be expected from these considerations. The fact that Franck
and Hertz and McLennan obtain single-line spectra is due, as Millikan
also pointed out, to their probably not having used dense electron streams.

It is now evident that the quantity which can be called the ionization
voltage of a gas or vapor is not necessarily the minimum voltage required
to ionize the gas or vapor. This latter voltage, we have seen, depends
more on extraneous conditions than on the nature of the substance, and
can therefore not be considered a property of the substance. According
to the views postulated above ionization voltage must be defined as
the equivalent of the minimum energy necessary to completely detach
an electron from the normal atom, and is therefore the least voltage
through which one electron must drop to ionize the normal atom. This
quantity is a property of the substance only and does not depend on
extraneous influences. It is determined by the equation

^ e '

where Ve is the convergence frequency of the principal series, h is Planck's
constant and e the elementary charge.

556 H. J. VAN DER BIJL.

.SBWIKSi

In view of the disturbing influences discussed above it would seem
that the experimental determination of the ionization voltage is not a
simple matter. The observed ionization at voltages below that required
by the Bohr theory does not necessarily invalidate this theory. On the
contrary, the fact that ionization is observed to start at the voltages
necessary to cause the minimum displacement of an electron from the
outermost orbit of the normal atom, seems, on the basis of the interpre-
tation given here, to lend support to the Bohr theory.

Rbsbarch Laboratory, Western Electric Co.,
New York,

May 30, 1917.

Nc^^'J ^^^ PARALLEL JET HIGH VACUUM PUMP. 557

THE PARALLEL JET HIGH VACUUM PUMP

By William W. Crawford.

EFFORTS to make a vapor aspirator-ejector produce a high vacuum
have met the difficulty that the jet, when surrounded by a high
vacuum, disperses and refuses to entrain the gas. Gaede^ overcame this
difficulty by confining the vapor stream in a practically continuous wall,
through a narrow slit in which the gas enters the vapor stream. Williams*
constricts the stream at the point of entrainment, causing it to pass this
point with an increased velocity and reduced pressure, and finds that
the narrow slit can be practically dispensed with. He provides a water-
cooled surface at the point of entrainment to condense the vapor which
tends to pass into the vacuum space. Langmuir* investigated the condi-
tions in this type of pump more thoroughly, and reached the conclusion
that the cooled surface is the essential element; the constriction of the
stream does not appear in Langmuir's pump. Langmuir holds the view
that a jet necessarily must disperse in a vacuum, and utilizes the dis-
persing vapor (apparently the major portion of the jet) to urge the gas
along a surface on which the vapor condenses, and into the remainder
of the jet, which delivers the gas to the rough pump.

According to the kinetic theory of gases, the paths of molecules
between collisions are substantially rectilinear. At very low pressures,
these paths become limited only by the walls of the chamber.* If the
molecules emanate from a point and condense upon the walls, the linear
path becomes evident from the location of shadow patterns of obstacles
in the chamber.* Wood* has shown by this method that a mercury
jet may be produced which does not disperse materially in a high vacuum.

If the molecules in Wood's jet retain variegated velocities, it seems
clear that the limiting density for the jet cannot be much higher than
that for Knudsen's molecular flow, since the faster and slower molecules

* Annalen der Physik, 19x5, p. 357.

* Phys. Rev., May, 1916, p. 583.

» Phys. Rev., July, 1916. p. 48, Jour. Frankl. Inst., Dec., 1916, General Electric Review,
Dec., 1916. See also Knipp, Phys. Rev.. April. 1917. p. 311, and Jones and Russell. Phys.
Rev., Sept., 1917, p. 301.

* Knudsen, Ann. d. Phys.. 4. 28, 1909. p. 75.
•Anthony, Trans. A. I. E. E., 11, 133, 1894.
•Phil. Mag., Aug., 1915.

558

WILLIAM W. CRAWFORD,

rSBCOND

LSbubs.

would have to traverse the entire length of the jet without colliding with
molecules which they are overtaking, or being overtaken by. Such a
jet could exert little mechanical effect on a gas, the molecules of which
would pass freely through the jet in any direction.

If, however, a jet can be produced in which the molecules are moving
not only in parallel directions, but also with nearly equal velocities,
then collisions should disappear between the vapor molecules, even if
the density of the vapor is far above the limit for ordinary molecular flow.
Moreover, if collisions do occur, the resultant velocities must also be
nearly equal and parallel, since^ only the direction of the relative velocity,
and not the velocity of the common center of gravity of the two molecules,
is altered by the collision. A gas molecule moving with the jet could
enter it readily, but would be effectively prevented from returning

Fig. 1.

Fig. 2.

against the jet by the fact that it then meets a relatively enormous
number of molecules, with some of which it must collide.

The line of thought indicated above led the author to try pumps of
the form illustrated in Figs. I and 2. In Pump No. 6 (Fig. i), the
vapor generated in the boiler 5 at a pressure of lo mm. of mercury or

» Maxwell. Scientific Papers, Vol. I. p. 377 et seq.

No"^*] ^^^ PARALLEL JET HIGH VACUUM PUMP. 559

more, escapes through the narrow throat 7*, which, it will be noted, is
at a considerable distance ahead of the point of entrainment. The
vapor expands in the diverging nozzle iV, and the issuing jet passes
through the tube £, which it fills, and condenses in D, mostly, it is found,
at the upper end. A slight amount of vapor escapes into the chamber
A, and condenses there. The condensed vapor drains back through the
tubes a and 6, to the boiler. The gas to be pumped enters through C.
Pump No. 5, Fig 2, is similar except as to arrangement, and the omission
of the enlarged chamber A of Fig. i, where its function is partly to
condense the vapor arising from the mercury draining back from D.

The dimensions of the nozzle and passage E were about the same in
the two pumps, viz.: Throat, 0.24 cm., mouth, 1.3 cm. diameter, ratio of
areas, 30. Diameter of £, 2.5 cm., length, in No. 6, 5 cm., in No. 5
(measured from the end of iV), 2.5 cm.

This form of nozzle is a result of the application of the principles of
nozzle design used in steam engineering practice. As is well known, at
the point of minimum area the pressure is never less than about half
the initial pressure, the minimum pressure and maximum velocity occur
in the diverging passage beyond the constriction. The reason for this
difference from what takes place with a liquid lies in the expansive
nature of the vapor.

It is found that a jet produced in this manner disperses only slightly,
and will, if of a proper density, entrain the gas to be pumped even if the
pressure of the gas is not over a thousandth of the computed internal
pressure of the jet. No diffusion slit or condensing surface is necessary
at the point of entrainment, in fact, the tube E surrounding the jet may
be artificially heated to a point where no mercury can be seen condensing
on it, without sensibly impairing the action. To all appearances the
jet itself reentrains and expels most of the vapor which is diffusely re-
turned after striking the wall. The pumps are entirely air-cooled.

The theory of the formation of the jet appear^ to be that on account
of the high initial pressure and the cooling due to the great ratio of expan-
sion, the relative velocities of the molecules are much reduced, while they
all acquire a very great common velocity in the direction of the jet.
The absolute velocities are therefore nearly equal and parallel, as desired.

Certain properties of the jet in these pumps have been computed
approximately, on the assumption that the vapor remains saturated, and
n^lecting friction, with the following results: Initial pressure, 18 mm.,
final pressure, o.i mm. Temperatures, 200 and 81® C., respectively.
Velocity of jet, 42,000 cm. per sec. Relative molecular velocity, r.m.s.
value, 21,000 cm. per sec. Mean free path, relative to jet, 0.08 cm.

560 WILLIAM W. CRAWFORD, [^S2

If the free jet is i cm. long, each molecule is in it only 1/42,000 sec., in
which time a molecule having the mean velocity will travel 0.5 cm.
relative to the jet, and hence make only 6 collisions. Probably, due to
the suddenness of the expansion and the absence of nuclei, the vapor
enters the supersaturated condition, with a resultant lower temperature
and lower relative molecular velocity, than that stated.

The fact that frictional effects do not destroy the jet suggests the expla-
nation that the parallel motion reduces the number of collisions against
the walls, both in the low pressure part of the nozzle, and in the com-
pression passage £.

If the density of the jet exceeds a well-defined limit, the pump prac-
tically stops working, and the results approach those described by
Langmuir for an aspirator with a dispersing jet. The author believes
that this limit is established by the density of the dispersing fringe, which
is probably proportional to the density of the jet, and occurs at the point
where the mean free path of gas molecules entering the fringe becomes
less than the total depth of the fringe.

Tests.

In testing these pumps, the limitations of the apparatus available
were such as to preclude obtaining extreme vacua. Pressures were
measured on the intake side by a 500 c.c. McLeod gauge sealed to
the pump through about 60 cm. of i cm. glass tubing, and on the dis-
charge side by a 65 c.c. McLeod gauge, connected with glass tubing and
rubber joints. The glass was not treated in any way, and the rate of fall
of pressure, and the ultimate vacuum, were determined somewhat by
the evolution from the glass and the friction in the tubing. The results
were about the same with and without a drying agent (PiC)») in the
vacuum space, the McLeod guage readings did not show symptoms of
water vapor, indicating that the pump was effective in removing the
latter.

The speed of the pump (cubic centimeters of gas per second at intake
pressure) was determined by allowing air to leak in through a calibrated
orifice located in the intake tube at a distance of 10 cm. from the pump,
and was computed by the relation

5 = 745x '

1000

where p = pressure in vacuum space, millimeters of mercury,

g = rate of leakage, cu. mm. per sec., measured at 760 mm.
pressure.

NO.S. J

THE PARALLEL JET HIGH VACUUM PUMP.

561

The speed is therefore that of the pump in series with 10 cm. of tubing,
19 mm. in diameter.

Results.

Pump No.

Ob««nrfttioD
No.

Boiler Pros-
•ore. Mm.

Le«k«cc, Co.
Mm. per Sec.

PrtMorc, Mm.

Speed C.C.,

Discharfe.

Intake /.

114

117

102

108

0.84

1.76

3.14

2.0

1.5

1.6

1.8

2.2

1.6

1.8

1.25

1.76

1.78

0.014
0.017
0.090
0.035
0.055
0.027
0.024
0.020
0.019
0.025
0.022
0.030
0.019
0.013
0.019
0.019
0.057
0.080

0.00002

0.00004

0.00002

0.0061

0.00105

0.15

0.11

0.035

0.013

0.0025

0.0026

0.0021

0.0029

0.0058

0.0086

0.017

0.00005

O.OOOl

105
1270

16.5

31.5

93
550
640
580
470
164
155

Observations 6 to 16 inclusive are plotted in Fig. 3.

Pump No. 4 resembled No. 6 except that the nozzle had half the linear

Boiler Prtssu re, mm.

Fig. 3.

562 WILLIAM W. CRAWFORD, [toSS

dimensions, and the chamber A was narrow, becoming heated by con-
duction and radiation from the vapor tube, and forcing the retrograde
vapor to pass up into C before condensing. This was the condition in
reading No. 4. A slight amount of cooling on A had the effect of greatly
increasing the speed (reading 5). Cooling E as well as A gave little
further increase in speed (perhaps 20 per cent.). Decreasing the radia-
tion from E by wrapping it in cotton wool caused the mercury previously
condensed on the wall to slowly reevaporate, but while this was occurring
the speed was not sensibly reduced. In readings i to 3 the pump was
not cooled.

In all the readings given for No. 5, the passage E was electrically
heated and no visible condensation took place in it. Even without this
heating, relatively little mercury condenses here, but it was desired to
prove that condensation was not a factor in the entrainment. These
readings show clearly the necessity of the jet density lying within well-
defined but not necessarily narrow limits. The ratio of expansion being
fixed by the nozzle, the density of the jet may be considered as propor-
tional to the boiler pressure. Simultaneously with the reduction of the
speed by high boiler pressure, retrograde vapor could be seen condensing
rapidly in C. The effect of too low density is also shown.

Only meager tests were made on No. 6, which was connected to the
apparatus through narrow rubber tubing, preventing a favorable deter-
mination of the speed. This pump seemed to produce good results with
a slightly higher boiler pressure than No. 5, indicating the usefulness of
chamber A in rarefying and condensing the retrograde vapor, permitting
the gas to be entrained despite the higher jet density.

The following comparison of the speed of these pumps with the values
stated by Langmuir for the condensation pump is of interest.

Style.

Condensation

Parallel jet No. 5 .
Parallel jet No. 4.
Condensation . . . ,

Dimmcttr, Cm.

Speed, C.C. per Sec.

Ratio Speed/Dlam.i.

2.0

200

2.5

500

2.5

1200

192

7.0

3000

It is probable that the jets in Nos. 4 and 5 are not the best that can
be produced by this method, and that with slightly different proportions,
greater speeds can be obtained with tubing of the same size. The speed
of the intake passages of No. 5, computed by Knudsen's formula, is
about 5,000 c.c. per sec., the difference is to be attributed to the resistance
of the jet to the entrainment of the gas.

Nos?^*] ^^^ PARALLEL JET HIGH VACUUM PUMP. 563

The work reported herein was done partly at the local laboratory of
the Victor Electric Corporation, of Chicago, and partly at the University
of Pennsylvania. The author wishes to express his thanks to Mr. H.
Clyde Snook, and to Dr. Harold Pender, for suggestions, encouragement,
and facilities.

Dbpartmbnt op Electrical Enginbbring,
Towns Scientific School,

University of Pennsylvania.

564 PAUL T. WEEKS.

A DETERMINATION OF THE EFFICIENCY OF PRODUCTION

OF X-RAYS.

By Paul T. Wbbks.

THE purpose of this investigation was to measure the energy of the
X-rays emitted by a Coolidge tube by means of their heating
effect; to determine the efficiency of production of X-rays, i. e., the ratio
between the energy of the X-rays and the energy supplied to the X-ray
tube; and to determine the variation of this efficiency with the potential
across the tube.

A variety of methods have been used for the measurement of X-ray
energy.

The first measurement was made by Dorn^ in 1897 by means of a
differential air thermometer. A bolometer method was used by Sch6ps*
in 1899, by Rutherford and McClung* in 1900, by Wien* in 1905, by
Angerer* and Carter* in 1906. Bumstead^ in 1906 measured the energy
by means of a radiometer and Adams' in 1907 used a radiomicrometer.
A thermopile was employed by Wien* and by Hoepner* in 1915. In
several cases the energy supplied to the tube was not determined so that
no conclusions could be drawn as to the efficiency of production of the
X-rays. Wien/ Angerer,* and Carter,* however, measured the energy
carried by the cathode rays and determined the value of the efficiency.
Carter also determined the variation of the efficiency over a considerable
range of voltage.

The ionization produced by X-rays has also been used as a means of
determining the efficiency of production of the X-rays. Rutherford
and McClung* early found a value for the energy required to produce
an ion in air by X-rays. In 1913 Beatty^® determined the number of
ions produced by the total absorption of X-rays. From the work of

» Wied. Ann., 63: 160.

* Dissertation, Halle.
» Proc. Roy. Soc., 67: 245.
< Ann. d. Phys., 18: 991.

* Ann. d. Phys., 21: 87.

I •Ann. d. Phys.. 21: 955.

j 7 Phil. Mag., 11: 292.

I • Proc. Am. Acad., 42: 671.

•Ann. d. Phys., 46: 577.

10 Proc. Roy. Soc., 89: 314.

Vol. X.l
No. 5. J

EFFICIENCY OF PRODUCTION OP X-RAYS.

565

Others he computed the total number of ions which would have been
produced directly by the cathode rays which excited the X-rays. The
ratio of these two quantities he took as the efficiency of transformation
of energy from cathode rays to X-rays. He gives the following relation

X-ray energy

as the result of his work:

= 5.1 X io~*i4/3*, where

cathode ray energy
A is the atomic weight of the metal of the anode and fi is the ratio of
the velocity of the cathode rays to the velocity of light. In 1912 Eve
and Day^ determined the total number of ions produced in air by X-rays
and found a value for the efficiency of production of X-rays from the
energy supplied to the tube and the energy required to produce an ion,
as determined from other exj)eriments. Recently, 191 5, Rutherford
and Barnes* have made a determination of the energy output of a
Coolidge tube from the total number of ions produced and the energy
required to produce an ion by alpha rays. The energy supplied to the
tube was measured and from this the efficiency computed.

Below is given a summary of the results of previous work. The values
of the efficiency given are computed for the total energy which would
appear on the outside of the tube on the supposition that the energy is
emitted equally in all directions throughout a whole sphere.

Obsenrcr.

Wien

Angerer

Carter

Hoepner

Eve & Day

Rutherford and Barnes
Beatty

Mtthod.

Bolometer

Thermopile

Bolometer

Thermopile

Ionization

(Abs. only by thin Al.)

PottDtiml.

58.7 K.V.
58.7
Low
59
65

(11 cm. gap)
48
48
59

Efficiency.

.00143

.00183

.0004

.00062

.00029

.0001

.00059

.0019

.0023

In view of the diflferences in the values obtained by the various
observers by means of the heating eflfect it seemed to be desirable to
make a new determination under the more favorable conditions of better
control of current and potential and larger power input made possible
by the Coolidge tube.

Description of Apparatus.

A bolometer method was used, one of two similar resistances being
exposed to the X-rays and the relative change in its resistance caused

»Phil. Mag., 33: 683.
* Phil. Mag., 30: 361.

566

PAUL T. WEEKS.

[

Sboomd

^J?^JJ?JJ//I/?//7?/J///?/////I//7\

y^i/i/}^^

2ZZZZZ2ZZZZZZZZZZZZZZI.

Caf^boonl ^^LiCfd ^Wood ^

Tin

Fig. 1.

by the heating effect detected by means of a Wheatstone's bridge and
a galvanometer. The resistances were made of .056 mm. lead foil cut
in grid form and folded back and forth on itself so as to form a continuous
screen of about i mm. thickness which would absorb almost completely
the incident radiation. Thin paper was used for the insulation between
layers. The resistance of the grid which was exposed to the X-rays was
4.35 ohms and of the other 3.87 ohms. To protect the resistances from
fluctuations in room temperature they were enclosed in a Dewar cylinder

as shown in Fig. i. The resistance
to be exposed, called A^ was placed
in front of the comparison resistance,
B. Between them was placed a 2
mm. lead screen and in front of A
a similar screen with an opening 6.5
by 6.45 cm. This was 29.4 cm. from
the target, so that it subtended
/251.5 of the whole sphere. The end of the Dewar was closed with a
cardboard .85 mm. thick. The Dewar was enclosed in a wood box and
the end packed with wool to reduce the conduction of heat.

A D'Arsonval galvanometer was used, of the Leeds and Northrup
high voltage sensitivity type. This was connected with a shunt so as
to be very nearly critically damped. With a measuring current of .07
ampere a change of one thousandth part in the bridge ratio gave a
deflection of 250 cm. at a scale distance of 4.8 meters. The entire
bridge circuit was enclosed in a grounded metal cage to prevent inductive
disturbances. The part of the cage in the path of the beam of X-rays
was formed by a sheet of aluminum .09 mm. thick, which served also to
cut off all direct heat radiation from the tube.

High potential uni-directional current was secured by means of a
closed core transformer and mechanical rectifier. The tube current was
measured by means of a D.-C. milliammeter. The filament of the
X-ray tube was heated by means of a lead storage battery. For measur-
ing the potential across the tube a sphere gap was first used. This was
found to be unsatisfactory for measurements during the course of a run
but served for calibrating. The sphere gap consisted of two brass
spheres, each 6.5 cm. in diameter, placed horizontally. Each of these
was connected to the line through a resistance of distilled water, the total
resistance in series with the gap being of the order of 10 m^ohms. A
tendency for the potential to rise to an abnormally high value before
spark-over would occur was almost entirely eliminated by placing a
tube containing some radium bromide close to the gap. For indicating

vSx^'] EFFICIENCY OF PRODUCTION OF X-RAYS, 567

the potential during the course of a run a balance form of electrostatic
voltmeter was constructed. The movable part of this, a 4 cm. sphere,
was suspended from a spiral spring and was immersed in oil above a
flat metal plate. This voltmeter was easily read and was found to follow
small changes in the potential with practically no lag.

Method of Observation.

Due to the inequality of the two resistances the measuring current
produced a change in the resistance ratio. In addition there was unequal
heating in the two from the stray heat conducted in from the outside.
Consequently there was a continual drift of the spot of light. However
this was not erratic and the rate was determined before each exposure
by taking four or five readings at one-minute intervals. The rate of
drift could be kept within the desired limits by varying the room tem-
perature slightiy. The time between exposures was ordinarily eight to
ten minutes as it was necessary to wait for the target to cool as well
as to determine the rate of drift.

Exposures were made for 30 seconds, the current and potential being
held nearly constant during this time. In Fig. 2 is given a curve showing
the galvanometer deflections during a typical ex(X)sure, the circles in-
dicating the readings taken. Some corrections were necessary in deter-
mining from these readings the actual rate of heating due to the absorbed
X-rays. By taking account of the rate of drift it was possible to deter-
mine the deflection due to the absorbed X-rays only. Then from the
observed rate of cooling during the first 15 seconds after the exposure
it was possible to correct for the cooling which took place during the
exposure. Thus in the case shown in Fig. 2 the deflection due to the
X-rays only was taken to be 6.90 cm. at the end of the exposure and 5.96
cm. 15 seconds later. These values give 15.95 cm. per minute for the
rate of deflection due to the X-rays alone. From this rate of deflection
and the heat sensibility of the bolometer, determined later, the amount
of energy absorbed could be computed in joules per ampere-second of
tube current. This multiplied by 251.5 gives the energy for the whole
sphere.

A set of 5 to 10 exposures was made at one current and potential and
then the tube adjustment changed or a different absorbing screen inserted
in the path of the rays and a similar set obtained. For each potential
several such sets were made on different days and with different values
of tube current. Readings were taken at eleven different potentials
and at four of these readings were taken with four different thicknesses
of aluminum in the path of the rays for the purpose of determining the

568 PAUL T, WEEKS. [

absorption curves. The results for each potential were derived from

at least 20 and in some cases 40 separate exposures. For the absorption

curves only from 10 to 20 exposures were made with each screen at each

potential.

Calibration of the Bolometer.

The heat sensibility of the bolometer was determined by sending
through the resistance A a known current for 30-second intervals and
observing the resulting galvanometer deflections. While the heating
current was flowing the measuring circuit was kept open. Readings were
taken similar to those taken in the determination of the heating due to
the X-rays and similar corrections were made. The heat produced in
resistance A was computed from its resistance and the current flowing
in it. Since A was in parallel with part of the bridge resistance the
observed value of the current had to be corrected for the small current
which flowed through the bridge. The mean sensibility obtained was
50 cm. per joule per .07 ampere measuring current.

Measurement of the Energy Supplied to the Tube.

If the tube had been operated by steady direct current it would have
been sufficient to take the product of the tube current and the potential
across the tube as the power supplied to the tube. However in the
case of the rectified alternating curreht the current and voltage were
both pulsating and the wave form of neither was known. Furthermore
the voltmeter deflections were determined by the root mean square
value of the potential and the milliammeter deflections by the mean
value of the current. Therefore it was thought advisable to make a
direct determination of the power by means of the heating effect.

The method employed was to immerse the tube in an oil bath and
measure the energy supplied by means of the rise in temperature of the
oil. The tank was made of tin and was enclosed in a wood box. It was
just large enough to contain the tube and allow of sufficient insulation.
Kerosene oil was used for the bath. The tube was covered over with
a black insulating cloth which was also immersed in the oil. A small
propeller driven by a motor served to keep the oil well stirred. The rise
in temperature was indicated by means of two copper-advance thermo-
couples, two junctions being placed in different parts of the tank and two
in a container of oil outside. The thermo junctions were connected in
series with the galvanometer used in the previous measurements and with
950 ohms resistance and gave about 17 cm. deflection per i** C.

Continuous runs were made, the potential and current being kept as
nearly constant as possible and galvanometer deflections being noted

JJJJ-^] EFFICIENCY OF PRODUCTION OF X-RAYS. 569

every minute. The duration of each run was from 15 to 20 minutes
and the rate of heating between 4 and 5 cm. per minute. The rate of
cooling was found before and after each run and the mean added as a
correction to the observed rate of heating. Several runs were made
and gave concordant results. The tube potential was kept at 31.6 K.V.
and the current at 4.80 m.-a. It was intended to make runs at other
potentials, but at this point the tube developed a leak and could not be
used furtlier.

Observations were next made on the heating produced by sending
current through a heating coil placed in the bottom of the tank. Condi-
tions were kept as nearly as possible the same as before, the stirring
device being kept in operation and the tube filament lighted. The
current through the coil and the potential difference across its terminals
were measured at intervals throughout the run. Runs were made with
the power adjusted to give heating at rate3 somewhat above and some-
what below that produced by the operation of the tube. These gave 188
watts as the power corresponding to a tube potential of 31.6 K.V. and a
tube current of 4.80 m.-a.

Calibration of the Voltmeter.

The voltmeter was calibrated by means of the spark gap and an
electrostatic balance. This latter could be used only for the lower
voltages on account of spark-over. It was found that a given voltmeter
reading corresponded to a 20 per cent, higher R.M.S. potential, as deter-
mined by the balance, with rectified current than with alternating cur-
rent. The difference was somewhat larger according to the spark gap
readings but approached the same value at higher potentials. The
oscillations introduced by the rectifier were probably responsible for the
lower spark-over potentials with the rectified current. The differences
in the voltmeter readings with alternating and rectified current were
undoubtedly due to a leakage of charge over the surface of the glass jar
containing the oil in which the attracted sphere was immersed. This
would change the distribution of the field and so change the vertical
force on the attracted sphere. A comparison of alternating potentials
as determined by the electrostatic balance and by the spark gap indicated
that the transformer used gave a wave form having a peak value 7 per
cent, higher than a sine wave of the same R.M.S. value. The spark
gap potentials used were those given by Peek^ for 6.25 cm. spheres.
The final calibration of the voltmeter was then determined from the

"••Dielectric Phenomena," by F. W. Peek; same in A. I. E. E. Standardization Rules.
.1915.

570

PAUL T. WEEKS.

alternating current spark-over voltages with the corrections indicated

above.

Computation of Efficiencies.

In the experiment to determine the power supplied to the tube it was
found that with a potential of 31.6 K.V. and 4.80 m.-a. current the power
supplied was 188 watts. The product of kilovolts and milliamperes is
152. This gives a correction factor of 1.24. Inasmuch as the milliam-
meter read mean values of the current a rectified sine wave voltage and
current would have given a factor larger than unity, but the large value
found under the conditions of the experiment is surprising. It would
have been desirable to determine this factor for other voltages. The
efficiency of production of the X-rays was taken to be the ratio of the
total number of joules of X-ray energy given out per ampere-second to
the number of watts supplied per ampere. The values found for the
efficiency are given in the fourth column of Table I.

Table I.

K. v.

28.3
31.6
34.5
37.1
39.5
41.6
43.5
45.2
48.2
50.8
53.9

Input
WatU

Amp.*

35.1 X 10»

39.2

42.8

46.0

48.9

51.5

53.9

56.1

59.8

63.0

66.8

Ob«. X-Ray
Bnergy

Joules

Amp.-sec*

17.1
23.4
30.7
38.3
46.3
52.3
59.3
66.9
84.3
93.6
118.0

Bfflciencj

without

Correction.

Abeorp-

tion
Factor.

0.49 X lO-i

1.20

0.60

1.17

0.72

1.15

0.83

1.13

0.95

1.11

1.02

1.10

1.09

1.20

1.08

1.41

1.07

1.49

1.07

1.77

1.06

Total Energy

Jonlea

Amp.-Sec*

20.5
27.4
35.3
43.2
51.3
57.5
64.6
72.3
90.2
100.2
125.1

Total
Bfflciency.

0.58 X 10-»

0.70

0.82

0.94

1.05

1.15

1.20

1.29

1.51

1.59

1.87

Correction for Absorption.

To determine the correction for
the absorption in the screens which
were in the path of the X-rays ab-
sorption curves were obtained for
four different potentials. The re-
sults are shown in Fig. 3, Curves A ,
B, Cand D corresponding to R.M.S.
potentials of 50.8, 45.2, 39.5, and
31.6 K.V. respectively. These are
plotted as percentage transmission

%5 fni/t

Nas. J

EFFICIENCY OF PRODUCTION OF X-RAYS.

571

against thickness of aluminum in millimeters. In the path of the rays was
an aluminum screen .09 mm. thick and a cardboard screen .85 mm. thick.
This latter was estimated to be equivalent to .08 mm. of aluminum.
In Fig. 3 a line is drawn at the left of the axis at a distance corresponding

.^ /.O 15 2.0 ^.5 ^,omm,
Th/ckne^ Of Aluminum

Fig. 3.

to .17 mm. and the absorption curves are continued to intersect this.

The intercepts on this line give the correction factors for determining

the total efficiency. The correction factors for intermediate potentials

were found by interpolation. These values are given in the fifth column

of Table I. In the seventh column are given the values of the total

efficiency for the total energy outside of the tube. It is evident that no

accurate estimate can be made of the energy absorbed in the walls

of the tube.

Discussion of Results.

In Curve A of Fig. 4 is shown the variation of the efficiency, un-
corrected for absorption, with the potential. In Curve B is shown the
variation of the corrected efficiency with the potential. The shape of
these curves is similar to that given by Carter.* It seems very possible
that if correction for absorption in the tube could be made the efficiency
might be found proportional to the potential as required by Beatty's
formula (see p. 2). In Fig. 5 the efficiency is plotted against kilovolts
squared, Curve A giving uncorrected values and Curve B the values
corrected for absorption. From these it would seem that the X-ray
energy emitted through the tube is nearly proportional to the cube of the
potential. The fact that the photographic effect is proportional to the
square of the potential, as found in practice, may be explained on the

572

PAUL T. WEEKS.

ground that the harder rays are absorbed to a less extent in the photo-
graphic emulsion and are also less effective because their wave-lengths
are much less than those of the characteristic radiations of bromine and
silver.
A comparison of the results given here with those obtained by other

ley-

no 30 -^ Jo 60 A", k"
Pofenh'a/ Ckm.^.)

Fig. 4.

observers using the heating effect shows that the values of the efficiency
are higher in general than those previously given. According to Ruther-
ford and Barnes" Beatty's values should be divided by a factor of 2 or
3 for comparison with the other values given because in his experiment

e^j^'w

IJB

v,/^

iooo Zooo

Fig. 5.

3eao/cv?

the X-rays did not pass through the glass wall of the tube but only
through a thin aluminum window. Both Carter and Wien used an in-
duction coil and measured the potential by means of a spark gap which
would give very nearly peak values. It would seem that their values for

Na"^'] EFFICIENCY OF PRODUCTION OF X-RAYS. 573

59 K.V. should be compared with a value given here for a potential
between 40 and 50 K.V. On this assumption the results given here
are somewhat below the corresponding results given by Wien and con-
siderably above those given by Carter.

The values of the efficiency found are quite different from those found
by ionization methods. The results given by Rutherford and Barnes
are based on a value for the energy required to produce an ion which
was determined from measurements with alpha rays. This was taken
to be the same as the energy required to produce an ion by means of
X-rays, A similar assumption is involved in Beatty's results, namely,
that the same amount of energy is required to produce an ion by means
of cathode rays as by X-rays. The work of Barkla and Philpot,^ of
Wilson* and of others' would indicate that ionization by X-rays takes
place through the intermediate production of high-speed electrons.
Hence the assumption made by Beatty seems justified if all the energy
of the X-rays is given to these electrons. This has not been proven
definitely. The work of Kleeman* indicates approximate proportionality
between the ionization produced in different gases by alpha, beta and
gamma rays. The work of Rutherford and Robinson* showed that in
the case of alpha and gamma rays from radium C the heating and ioniza-
tion were nearly proportional, although there were large errors involved.
If this proportionality is accepted as established it would seem to justify
the assumption made by Rutherford and Barnes in their determination
of the efficiency of production of X-rays.

However if we compare the values for the energy required to produce
an ion as found in this way and the values found by other means there
are seen to be large discrepancies. A recent determination by Bishop*
gives 1.67 X lO"^® ergs per ion, or one third of that given by Rutherford
and Barnes, 5.1 X io~" ergs (33 volts). Using this value for the energy
to produce an ion Rutherford's and Barnes's value for the efficiency of
production of X-rays becomes .2 X lO"* for 48 K.k. The value given
by Eve and Day is based on an energy of 2 X io~^^ ergs per ion and is
thus seen to be in fair accord with this as they used a somewhat lower
potential. The value for ionizing energy obtained by Rutherford and
McClung,' when corrected for the large value for "e" used by them,
is 1.4 X io~" ergs per ion. The work of Rutherford and Barnes shows

iPhil. Mag., 35: 832.

« Proc. Roy. Soc., 87: 277.

* Bragg. "Studies in Radioactivity." Chapter 12. See also (14).

* Phil. Mag.. 14: 618.
•Phil. Mag.. 25: 312.
•Phys. Rev.. 33: 325-

574 PAUL T. WEEKS. [

that at 48 K.V. there would be produced by a Coolidge tube 12 X 10^

ions per second per watt input. Using the value of efficiency found in

the present investigation this would correspond to an X-ray energy of

1.5 X 10* ergs per second or 1.25 X io~*® ergs per ion, a value nearly

the same as that found by Rutherford and McClung by a similar method.

From these four investigations it would appear that the energy of

the ions produced by X-rays is only a fraction of that emitted from the

tube in the form of X-rays. To a less extent the same thing is true of

alpha rays which appear to be more efficient than X-rays in producing

ionization. These conclusions seem to contradict the evidence given by

experiments with radioactive materials. To decide the point the heating

effect and the total ionization should be determined simultaneously or

under the same conditions.

Summary.

The energy given out in the form of X-rays by a Coolidge tube has
been determined by means of a bolometer. The values found lie between
20 and 125 joules per ampere-second for potentials between 28 and
54 K.V.

The energy supplied to the X-ray tube has been measured by its
heating effect.

The ratio between the X-ray energy and the energy supplied to the
tube, or the efficiency of production of the X-rays, has been found for
these potentials. This ratio varies between 0.58 and 1.87 X io~'.

The X-ray energy is found to be nearly proportional to the cube of
the potential across the tube.

A comparison of these results with those obtained by others on the
total ionization produced by X-rays indicates that only a fraction of
the energy of the X-rays is transformed into the energy of the ions
produced on total absorption in air.

I wish to express my indebtedness to Professor J. S. Shearer and to
other members of this department for suggestions and help given me
and for the apparatus put at my disposal.

Cornell University.
June. 191 7.

JJj^jf] WAVE-LENGTH OF LIGHT, 575

THE WAVE-LENGTH OF LIGHT FROM THE SPARK WHICH
EXCITES FLUORESCENCE IN NITROGEN.

By Charles F. Meyer.

IT was shown by Professor Wood^ in 1910, that radiations from the
aluminium or copper spark are capable of exciting ultra-violet
fluorescence in air and other gases, notably in nitrogen. The spark was
passed between a rod serving as lower terminal, and a plate with a
small hole in it serving as upper terminal, the spark striking just at the
edge of the hole (Fig. i). All the radiations from the spark, including
the visible and ordinary ultra-violet, pass up
through the hole in the plate, and if any of the
radiations cause fluorescence this can be photo- ^^
graphed by placing a camera off to one side of the
opening. The luminosity was first photographed
with a camera fitted with a quartz lens, and then
with a quartz spectrograph which had had its slit Fig. 1.

removed, the fluorescent jet itself serving as slit.

It was shown that the fluorescence in air consisted of the water bands
3064 and 281 1, principally the former. When an atmosphere of nitrogen
was used above the plate the water band 3064 appeared, and also the
three nitrogen bands 3369, 3556 and 3778.

Inquiring now into the origin of the luminosity which exists above the
opening, it is evident that the luminosity cannot arise from the simple
scattering of light from the spark by particles of dust, or by the air
molecules, as the spectrum of the luminosity is quite distinct from that
of the spark, and a quartz plate several millimeters thick, placed over
the hole, causes the light to disappear entirely.

That the fluorescence is excited by light in the Schumann or ultra-
Schumann region was the first hypothesis proposed by Professor Wood,,
and in some work by Wood & Hemsalech^ it was shown that the radia-
tions were transmitted, but only to a slight degree, by a thin piece of
clear fluorite. This fact indicated that the radiations exciting the
fluorescence lay beyond the Schumann region, for the limit of the

> Phil. Mag. (6), Vol. 20, p. 707.

* Phil. Mag. (6), Vol. 27, p. 899, 1914.

576 CHARLES P. MEYER, [^

ilCD

iKRIBS.

Schumann region is set by the transmission of fluorite in sufficient
thicknesses to form prisms and lenses.

Wood & Hemsalech also showed that the radiations exciting the
nitrogen bands were transmitted to a slight degree through quartz,
and in some experiments which Professor Wood and I performed,^ we
refracted the radiations through the extreme edge of an especially ground
quartz prism, the path in the prism being probably less than a tenth of
a millimeter. The index of refraction for the radiations came out
1.75 =b .08, that is, with a probable value greater than that of quartz
at the more refrangible end of the ordinary ultra-violet. This value of
the index of refraction made it seem as though the radiations were still
on the long wave-length side of the quartz absorption band, and possibly
not far beyond the limit of the Schumann region, if indeed they were
beyond it at all. Unless it should be that they were on the long wave-
length side of an entirely hypothetical second absorption band of quartz
well beyond the Schumann region.

At this time we also succeeded in reflecting the radiations from silicon
and speculum. The reflection experiment put me in mind of a method
by which the wave-length of all or a part of the radiations might be
approximately determined, in case they had a wave-length of seven or
eight hundred angstroms or more.

The method is as follows: A very fine slot (.2x2x4 ni^^ ) ^ cut in
a copper rivet, copper now being used for the spark terminals throughout.
Beneath this is fastened a small fragment of a grating with the lines
horizontal and parallel to the edges of the slot. . The rivet serves as one
spark terminal, a copper rod or wire as the other. Fig. 2 represents this
arrangement in vertical section, R is the rivet with the slot in it. The
rivet is driven through the thin copper plate C S is the rod serving
as the other spark terminal. G is the small fragment of a grating, the
lines being perpendicular to the plane of the drawing. The plate C
is bent into a right angle and fastened to an arm A which rotates on a
horizontal axis. This rotation allows the plate, the rivet, and the grating,
which are all fastened rigidly together, to be raised for the purpose of
cleaning the grating. Below the grating and the lower end of the plate
C is a thin horizontal plate with a rectangular hole in it 3 mm. by 4 mm.
This second plate forms the top of a metal box into which nitrogen is
introduced. The box also serves to keep out stray light and thus yield
a dark background. The apparatus is represented in perspective in
Fig. 3. The lettering of the parts is the same as in Fig. 2, only in addition
attention is now called to the representation of the upper portion of the

* Phil. Mag. (6). Vol. 30. 191 5-

VOL.X.

No.

X-l
5. J

WAVE-LENGTH OF LIGHT.

577

box B, with'^the'quartz window W, and some ten centimeters in front of
the window the lens L and the prism Pr^ and finally, off at a small angle,
the^photographic" plate P. The path of the central ray through the
optical system is indicated in the diagram. To avoid confusion in the
diagram, a metal tube of about 2 cm. diameter, through which nitrogen

Fig. 2.

Fig. 3.

is introduced into the rear of the box B, has been omitted from the
drawing. This tube also helps give an absolutely dark background,
which is essential. None of the drawings are to scale, some dimensions
being exaggerated for the sake of clearness.

Referring again to Fig. 2, it is seen that the beam of light, which
comes through the slot in the rivet, strikes the grating at a large angle of
incidence (73 deg.). The directly reflected beam comes off in the direc-
tion D in the figure. The first and second order diffracted beams come
off in the directions indicated. The exciting light causes fluorescence
along these three paths, and this fluorescence is photographed by the
optical system represented in Fig. 3. The object of the 30 deg. prism Pr,
is only to disperse the fluorescent light slightly and thus have a check on
the nature of the spectrum, and to be certain that there is no scattered
light. The prism is set for minimum deviation.

The dispersion caused by the prism would create confusion in the
directly reflected and the diffracted beams if the fluorescent spectrum
were a complicated one. But by using commercial nitrogen slightly
moistened it is possible to limit the fluorescence to the water band 3064
alone. Moist nitrogen was used in the experiments.

The first experiment of this kind was made over a year ago, with a

578 CHARLES F. MEYER. [^22

grating of 15,000 lines to the inch. But the diffracting angle was not
very great with this and it was felt that the experiment should be
repeated. This has recently been done, using a grating of 30,000 lines
to the inch. The results are much better, both on account of the more
suitable grating and on account of improved technique. Measurements
made upon two plates yield a result of 1300 A., with a probable error
of some 50 or 70 A. This is for the wave-length of the light exciting the
fluorescence of the water band 3064. The measures of course do not
admit of much accuracy, as the beams of fluorescent light are not sharply
bounded and not as intense as might be desired. The measures are made
on the first order beam. An exposure of six or seven hours is necessary,
the exposure being interrupted every half hour for cleaning the grating.
The best plate shows also the second order diffracted beam quite un-
mistakably.

The experiment with the fluorescence in nitrogen was checked by
placing a piece of cardboard in the chamber below the grating, the
cardboard lying in the plane of the drawing (Fig. 2). The light scattered
by the cardboard was photographed through the same optical system,
but passing in addition through a film of silver on quartz which allowed
only light of the wave-length of the silver transmission band to |>ass.
The wave-length calculated in this manner for the transmission band of
silver came out quite within the limit of experimental error.

As a result of the entire set of experiments it seems fairly certain
that at any rate some of the radiations previously studied lie at the border
of the Schumann region, and not beyond it. But it is difficult to account
for the rather low transparency of good fluorite for the radiations, and
the transparency of the air seems greater than would be expected. No
actual measurements have been made of the transparency of the air.
Professor Wood suggests that the fluorescence may be excited simul-
taneously by the radiations whose wave-length I have measured and by
some of shorter wave-length which are not readily transmitted by
fluorite or reflected by speculum metal. This may well be true.

It seems worth while, also, to consider the supposition that the radia-
tions whose wave-length was measured are the only ones causing fluores-
cence. In this case it is necessary to suppose that the transparency of
the fluorite is lessened, due to its proximity to the spark and consequent
heating, and there remains to investigate the transparency of the air.
It was hoped to do this, and to see whether there might possibly be an
effect of the spark upon the transparency of the air in its immediate
vicinity, but it has become necessary to lay the work aside for an in-
definite period.

JJJJ-^] WAVE-LENGTH OF LIGHT, 579

A rough estimate may be made from the reflection experiments of the
coefficient of reflection of speculum metal for the region near 1300 A.
It seems to lie between s and lo per cent, at 75 deg. incidence.

In conclusion, I wish to thank Professor Wood for making several
helpful suggestions.

Physical Laboratory,

University of Michigan,
June, 1917.

580 LEROY D. WELD AND JOHN C. STEINBERG. [toS

A STUDY OF APPARENT SPECIFIC VOLUME IN SOLUTION.

By LbRoy D. Wbld and John C. Steinberg.

IV. Results with Copper Sulphate.

TN a former paper/ of which this is a continuation, one of the authors
'^ has outlined the problem and explained the experimental methods
used, and presented the results obtained from experiments upon solutions
of potassium chlorate. For convenience, the matter may be summarized
as follows:

The apparent specific volume of the dissolved solute, designated by A,
is defined as the volume of the solution containing one gram of the solute,
minus the natural volume of the pure water entering into it at the same
temperature. The variations of A with temperature at constant con-
centration, and with concentration at constant temperature, in the case
of potassium chlorate, were set forth in the former paper, the experimental
arrangements being such as to follow with the most minute accuracy
changes in the density of the solution. The results gave strong indica-
tion of a minimum volume in solution, and the minimum was actually
reached in one super-saturated solution. This is strongly suggestive of
the behavior of water, pure or in mixture with dissolved substances, in
the neighborhood of freezing.

This research has been taken up by Mr. Steinberg and applied to
copper sulphate, with such modifications in procedure as have been
found necessary owing to the radically different properties of the sub-
stance under examination.

Copper sulphate is about three times as soluble as potassium chlorate,
and crystallizes with five molecules of water, whereas potassium chlorate
is anhydrous. The solutions of copper sulphate can be much further
super-saturated, that is, cooled much further below the saturation tem-
perature without crystallizing out, and then the crystals form very slowly,
growing in large masses at the bottom of the vessel, instead of suddenly
forming in the solution and settiing down like snowflakes, as do those of
potassium chlorate. The specific volume in solution, A, for copper
sulphate turns out to be very much less than for potassium chlorate,
and it decreases with increasing concentration in saturated solution.

» L. D. Weld. Phys. Rev., N. S.. Vol. 7, p. 421,

No. 5. J

APPARENT SPECIFIC VOLUME IN SOLUTION,

581

In preparing the stock solutions, small batches were at first made up
just before using, at approximately the concentration desired, and
specimens therefrom carefully analyzed to determine the exact strength.
Later an electrically controlled thermostat was constructed, in which

PiOt§

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OOSB

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NalO
. 0.3A6B

'enfp. 1

Fig. 7.

Fig. 8.

large flasks of stock solution could be kept at any desired temperature
for days, tightly stoppered and without crystallizing or evaporating.

The analyses were made electrolytically, the solution from which the
copper was deposited being stirred during electrolysis by the mechanical

Ad^

Ol<

<r4 ■

ol. /

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Fig. 9.

Fig. 10.

action of a strong magnetic field passing through the current-bearing
liquid.

Series of experiments made upon these solutions at several different
concentrations gave results which exhibit the minimum specific volume
in solution in the most striking manner, being far more pronounced than

582 LEROY D, WELD AND JOHN C. STEINBERG. \^Sm

any obtained previously with potassium chlorate. Four of these series

are shown graphically on the accompanying diagrams, which correspond

to the curves in Fig. 4 of the former section. Only the first of these,

that at concentration 0.24 (Fig. 7), was stopped too soon to exhibit the

actual minimum, whereas in the former research, only one was obtained

that did clearly exhibit it.

It is hardly necessary to enlarge upon the support which these later

results give to the theory previously suggested. The existence of the

minimum specific volume in solution can hardly be questioned. And it

further appears in this case, as in that of potassium chlorate, that the

temperature of this minimum is lower, the more foreign substance (water)

there is in mixture with the substance under examination (copper

sulphate). In fact, the phenomenon presents so many points of similarity

to those of the minimum volume of pure and solvent water as to lead one

to suspect very strongly that the phenomena are identical in nature,

whether to be explained on the Roentgen complex molecule hypothesis

or otherwise.^

Cob College,

Cedar Rapids, Iowa,
June, 191 7.

> See discussion at the bottom of p. 438 of the former paper, loc. dt.

VOL.X.1

ABSORPTION OP MERCURY VAPOR,

583

THE ABSORPTION OF MERCURY VAPOR BY TIN-CADMIUM

ALLOY.

By L. a. Welo.

THE writer has been engaged during the past year in a spectroscopic
study of the gases occluded in certain metals, and has had occasion
to keep the vapors from the mercury pump out of the evacuated tubes
by cooling the intervening connections in liquid air. An accident to
the liquid-air plant threatened to delay the work, as no gold leaf was at
hand to be used as a substitute, until it was suggested by Dr. L. T. Jones
that a portion of the connecting tubing be packed with some of his
supply of chips of a tin-cadmium alloy which is known to form an
amalgam with great ease. The alloy, commonly known as dentist's
amalgam, consists of two parts of tin to one of cadmium and was used
in the form of chips cut from a bar of the alloy with a milling machine.
They have a thickness of the order 0.05 mm., are 12 mm. in length and
vary in width from I to 3 mm.

A special test of the material has since been made to establish upper
and lower limits for the length of tubing to be packed. The apparatus
is shown in the figure. Eight branches are blown on centimeter tubing

'7b 6P£CT^06/f/IPH

To HroROosN %5uf^pL r

Fig. 1.

somewhat more than a meter in length and a Pliicker tube is attached
at the end. One branch leads to the pump and a second to the hydrogen
supply, this gas being chosen because its low atomic weight makes its

584 L. A. WELO. ggS

spectrum extremely sensitive to traces of mercury vapor.' The remaining
six branches are spaced as shown, rather than uniformly, in order that
fewer branches with fewer trials will determine the lower limit of length
in case it should be small. The PlQcker tube is equipped with aluminum
electrodes sealed in with platinum wire and is closed with a small right-
angled quartz prism instead of plate, that it may be placed between the
spectrograph slit and the'end of another tube already lined up and which
it was desired should not be disturbed.

The chips are closely'iHcked, without jamming, for a length of 96.8
cm. and the test consists in leaving mercury in successive branches,
banning at the pump end, for certain lengths of time and noting from
which branch mercury first appears as an impurity in the spectrum.
After the vapor had^ penetrated from the third branch the remaining

//Off

^«

6s
6i
Dist

Fig. 1.

branches were also used to see if there were any further increase in
strength of the mercury spectrum with time and with decrease of length
of packing. To make the test more severe the mercury is kept at a
temperature of 120-130 degrees with a small heating coil slipped over
the branch, giving vapor pressures of 0.7&-1.24 mm.'

Electrolytically prepared hydrogen dried with calcium chloride was
admitted from time to time and, after adjustment to a suitable pressure
indicated by the number of striations between the end of the capillary

:. 19. 105, 1904.

No!"s^l ABSORPTION OF MERCURY VAPOR. 585

and one of the electrodes, its spectrum was examined both visually and
photographically. The voltage on the exciting transformer, capacity,
length of spark gap and time of exposure were all constant. Nine of the
spectrograms appear in Fig. 2, where the numbers at the right refer to
the branch containing the mercury and the subscripts to the number of
hours it had been there. The two stronger hydrogen lines H^ and H^
could always be seen in the direct vision spectroscope, but the line Hy
is seen only in the first four spectrograms. After that it is suppressed
by the mercury which begins to enter from branch 3. The repeated
filling with hydrogen is seen to remove all of the carbon and cyanogen
and a part of the water vapor present as impurities before spectrogram
i-ioo is taken. It is seen that the very persistent line Hg 2537 is present
from the start, but that it also is removed on several fillings with hydrogen
until in spectrogram 2-42 it is all but visible. In the next, 3-4, where
the length of intervening packing is only 21.8 cm. the line again appears,
showing that the vapor is able to penetrate from the third branch. The
next three exposures show no progressive increase in intensity of the
many mercury lines with time. The last spectrogram was taken after
mercury had been distilled from the last branch to the vicinity of the
electrode.

The results may be summed up in the statement: A column of chips
of tin-cadmium alloy, which need not be more than 50 cm. in length,
is an effective bar to the passage of mercury vapor from the pump to the
vessel to be exhausted.

Physical Laboratory,

Univbrsity of California,
June 9, 191 7.

586 THE AMERICAN PHYSICAL SOCIETY. [

PROCEEDINGS

OF THE

American Physical Society.

Experimental Evidence for the Parson Magneton.

By L. O. Grondahl.*

THE Parson Magneton is an electron endowed with a magnetic moment.
This moment is estimated as being equal to 3.5 X io~" E.M.U.
Such an electron would be affected by a non-uniform magnetic field. A
conductor placed in such a field would therefore gain a negative potential
in that part which lies in the stronger portion of the field. The magnetons
would move into the stronger part of the field until equilibrium is established
between the magnetic and the electrostatic forces on the magnetons. The
potential so established in a conductor connected to earth and placed in a
field of 1,000 gausses would be 2.2 X 10"^ volts. This could be measured
by an electrometer.

A similar piece of evidence for the existence of the magneton is found in a
phenomenon with which the writer has done some work, namely, the effect
of a magnetic field on the thermoelectromotive force of magnetic substances.

If, for instance, a copper-iron couple is placed in a magnetic field, its E.M.F.
changes. This change, for which at present there does not seem to be a
satisfactory explanation, may be very simply explained in terms of the Parson
magneton.

Take a copper iron couple, the iron member of which is a short wire which
may be placed in a coil. If, while one junction is at o** C, the other at 100** C,
a current is turned on in the coil, there will be in general a stronger magnetic
field in the copper attached to the cold junction than in that attached to the
hot junction. At both junctions there will be an increase in magneton con-
centration in the copper due to the magnetic field. The increase in concen-
tration will, however, be greater at the cold junction, and if we accept this
explanation for thermal electromotive forces, this, since iron is positive with
reference to copper, means an increase in the thermal E.M.F.

Qualitatively this is in agreement with experimental facts as shown by
Fig. I. This is taken from an earlier paper. The ordinates represent change

^ Abstract of a paper presented at the meeting of the American Physical Society, December
27-30, 1 91 7.

VOL.X.1

Nas. J

THE AMERICAN PHYSICAL SOCIETY,

587

in E.M.F., the abscissae the strength of the field in which the iron is placed
It will be noticed that in the case of the iron we do get an increase in thermo-
electromotive force for all but the very low fields. It will be remembered,
however, that at low fields the permeability changes with temperature in the
opposite way, so the reverse effect would be expected. Nickel and cobalt
are negative with reference to copper, so the effect results in a decrease of the
electromotive force. There should be a reversal here, as well, since the
permeability varies in the same way. This is, however, not shown by the

^"^

£h.

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■■

■"

—

•

*~'

r-i

u 1

U "A

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bi*

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Fig. 1.

experimental results. These effects are so small that a reversal at low fields
might easily escape detection.

The quantitative relation is not exactly determined by experimental data
in existence at present. That it is of the right order of magnitude, however,
may be shown by the following: The change in permeability for iron in a
field of 100 gausses between o** and 100** C. is of the order of magnitude .7,
one specimen given in the Smithsonian tables changing from 177.9 at o** C.
to 177.2 at 100** C. The potential established by the field is H X 2.2 X io~^
volts. The difference in field intensity at the two ends may be something
like 30 gausses. Then the effect on the electromotive force would be 66 X io~^
volts. One specimen gives about 30, the other about 90 X io~^. In order to
expect a better check it would be necessary, of course, to determine both
quantities in the same specimen. It may be that the effect of the field varia-
tion in the magnetic member should be included in the calculation. This
would make the result larger.

The specimen of nickel also recorded in the Smithsonian tables, has a
permeability at o® C. of 1.6, at 100® C. of 1.5, with a field strength of 12,000

588 THE AMERICAN PHYSICAL SOCIETY. [

gausses. By the same method of calculation this will give a change in E.M.F.
due to the field of 600 X io~^ volts. Experiment gives as a maximum for
two specimens of nickel that were tried 450 X lO"^ volts. In the case of
cobalt, a similar calculation gives for a specimen recorded by Stiller in the
Physical Review, 49 X io~^ while for a specimen given in the Smithsonian
tables they get 96 X io~'. The specimen on which the experiment was tried
gave, approximately, 160 X lo"^ volts.

These calculations are all necessarily very rough, but nevertheless show
a fair quantitative agreement. In some experiments that are under way for
another purpose, the writer hopes to get the data necessary for a more accurate
determination of this quantitative relation. This explanation would involve
as a corollary that the copper would become magnetized due to the orientation
of the free magnetons. Assuming that the magnetons can be oriented and
that they would distribute themselves in all possible angles according to the
same law that Langevin assumed for paramagnetic substances, and assuming
that the number of free magnetons is the same as that usually assumed for
the number of free electrons in copper, it may be shown that for a field strength
of 9,000 gausses the intensity of magnetization of copper would be approxi-
mately 270. This may be masked by diamagnetism.

Carnbgib iNSTrruTB OF Technology,
January, 191 7.

S^/-] NEW BOOKS. 589

NEW BOOKS.

La Chimie des Elements RadioacUfs. By F. Soddy. Paris, Librairie Gauthier.
Villars, 1916. Pp. iv + 174. Price, 5 fr.

This is an excellent translation of Soddy's book, both parts being contained

in one volume. An improvement on the original has been made by putting

references in footnotes instead of at the end of the volume. As in the original,

an index for the second part is lacking.

E. P. L.

Croupes EXectrogenes en regime trouble. By L. Barbiluon. Paris, Gauthier-
Villars, 1915. Pp. ii + 306. Price, 11 fr.

This book is based on a course of lectures given in the Electrotechnical

Institute of the University of Grenoble. Its publication was delayed on

account of the fact that the author reported for military duty on the first day

of the war. It discusses from the theoretical standpoint the perturbations of

prime movers of all kinds and of direct and alternating current generators,

due to variations of velocity or of load, and the different methods of regulation.

Both analytical and graphical methods are employed, many of the latter

original. Characteristic gallic clearness of presentation and unusually good

typography combine to make this a very attractive volume.

E. P. L.

Ozone. By A. Vosmabr. New York: D. Van Nostrand Company, 1916.
Pp. xii + 197. Price, $2.50.

In the preface of this book, the full title of which is "Ozone, its Manufacture,
Properties, and Uses," we are told that the author's " main object was to give a
full outline of our personal experience and that same experience has made us
very critical about outside information." This emphasis upon his personal
experiences and opinions, and distrust of all ''outside information" is much in
evidence throughout the book. The volume is distinctly technical rather than
scientific; while it may be of interest and value to those engaged in the manu-
facture and use of ozone, the reader, must be alert for loose statements and

errors.

N. E. D.

The Dynamical Theory of Gases. By J. H. Jeans. Cambridge: University
Press, 1916. Pp. vi + 436. Price, $4.00.

A study of this second edition of Jeans* well-known treatise reveals many
improvements over the first edition of thirteen years ago. Much of the book

590 NEW BOOKS. I tow-

has been rewritten or rearranged so as to secure a more logical sequence in the
presentation. Many of the obvious steps formerly given in the mathematical
development have been omitted, thus securing a more concise treatment and
assisting the reader in obtaining a bird's eye view of the problem and its solu-
tion. The omission of some of the introductory and subsidiary matter found
scattered through the former edition likewise aids the reader. A number of the
tables have been greatly changed, some have been recomputed, others have
been based upon other authorities, some have been omitted, all give evidence of
a more careful selection of material. There is a noticeable decrease in the
number of references to and quotations of data from similar treatises and a
corresponding increase in the number of references to and of data derived from
original sources. An increase in the number of subheadings increases the value
of the volume as a book of reference. Much new matter has been added; the
former edition contained 420 numbered sections and 777 numbered equations,
the present edition contains 559 sections and 1044 equations.

The plan of the book is thus set forth by the author: " My primary aim in the
first edition of this book was to develop the Theory of Gases upon as exact a
mathematical basis as possible. This aim has not been forgotten in the prepar-
ation of a second edition, but has been combined with an attempt to make as
much of the book as possible intelligible to the non-mathematical reader. I
have adopted the plan, partially followed in the first edition, of dividing the
book to a large extent into mathematical and physical chapters." In taking
up a new book treating of the Dynamical Theory of Gases one at once looks to
see what prominence the author has given to the Quantum Theory and in what
manner he has introduced it. Jeans has "confined the Quantum Theory to the
last chapter; the difficulties arising out of the classical treatment have been
allowed to emerge in the earlier chapters, but have been left unsolved. The
last chapter merely indicates how these difficulties disappear in the light of
the new conceptions of the Quantum Theory; no attempt is made to give a full
or balanced view of the whole theory." The volume will be a valued addition
to the library of any one interested in the study of the dynamical theory of

gases.

N. E. D.

Second Series. December, 1917. Vol. X., No. 6

THE

PHYSICAL REVIEW.

UNIPOLAR INDUCTION AND ELECTRON THEORY.

By Gbo. B. Pegram.

THE most simply constructed apparatus for showing unipolar, homo-
polar or acyclic induction of electromotive force is a cylindrical
permanent bar magnet spinning about its axis with a stationary loop
of wire terminating in brushes which make contact with the rotating
magnet at two points, one nearer the end of the magnet than the other.
The old unipolar question is as to the seat of the electromotive force,
whether in the moving magnet or in the stationary wire; or, as some-
times put from the standpoint of the "cutting of lines of force" view,
does the magnetic field rotate with the magnet and by cutting the
stationary loop generate an electromotive force in it; or does it remain
stationary and cut the moving magnet? In terms of the electron theory
the question is whether electrons in the conducting material of the
magnet, aiid rotating with the magnet, are acted on by a force arising
from this rotation or whether it is the electrons in the stationary loop
of wire that are immediately influenced by the spinning of the magnet.
Of late certain questions involving the theory of relativity have also
been brought into discussions. The most recent articles on the subject
are by Bamett,^ Kennard* and Howe.*

While a very simply constructed apparatus for showing unipolar
induction results from using a cylindrical permanent magnet, a perma-
nent magnet is a complex thing, and for easier analysis we may well

^S. J. Barnett, Phys. Rbv., 35, 191a, p. 324 (a). 2, 19x3. p. 323; Phirs. Zeitz., 14, 1913.
p. 251.

* E. H. Kennard, Phys. Rbv. (2)» i, 1913. p. 355; (2). 7, 19x6. p. 399. Another article
by Dr. Kennard describing experimental results like those described in this paper, has ap-
peared in the Phil. Mag. for February, 191 7, but the theoretical treatment of Mr. Kennard
is so different from, the simple method of treatment herein attempted that I venture to
publish this as it stands and as it was presented before the American Physical Society in
October. 19x6. G. B. P.

* G. Howe, Electrician, LXXVI., p. 169. Nov. 5, 1915, and subsequent discussion.

591

592 CEO. B. PEGRAM, [

substitute a long helical solenoid carrying a steady current. In the end
the assimilation of a permanent magnet to a solenoid will be fairJy
obvious. Making use of such a rotating solenoid a unipolar induction
current can be obtained by having a conducting disc fastened coaxially
to the solenoid, and letting brushes from a stationary conducting loop
bear on this rotating disc at different distances from the axis of rotation.
The question then becomes in essence this: In which case will an electron
near a rotating long solenoid, with steady current through it, experience
a radial force, in case the electron is rotating with the solenoid as if
rigidly connected with it, or in case the electron is stationary?

Bamett, by studying open rather than closed circuits, that is, by
observing the displacement of charges on conductors in the field of a
rotating solenoid or magnet, made the first direct experimental attack
on the question, which has been followed up by Kennard. Bamett
used a condenser of concentric conducting cylinders with the outer
cylinder, closed at the ends, held coaxially in a solenoid which could
be magnetized and rotated. He found that the inside cylinder of the
condenser did not become charged if while the magnetized solenoid was
rotating a radial conductor made connection for a time between the
inner and outer cylinders of the condenser. This he proved by breaking
the connection between the inner and outer cylinders of the condenser
while the magnetized solenoid was rotating, stopping the current through
the solenoid, or bringing the solenoid to rest, and then testing the inner
cylinder for charge by connecting to an electrometer. Barnett varied
the experiment by arranging the cylindrical condenser coaxially with two
large round electromagnets, which with their magnetizing coils were
rotated in place of the solenoid. The result was the same as when the
solenoid was used, the inner cylinder did not become charged when a
radial connection was made between it and the outer cylinder. Both
these experiments therefore showed that. when the system which produces
a magnetic field symmetrical about an axis is rotating about that axis,
it does not establish an E.M.F. in a stationary conductor such as the
radial connecting wire between the two cylinders used in the experiment.
If the solenoid or magnet used in the experiment were kept stationary
and the condenser with the radial connection between the cylinders were
rotated, the inner cylinder would undoubtedly become charged. On
this point no one has raised any question, but Mr. Kennard has gone so
far with the experiment as to obtain observations showing the existence
of the charge on the inner cylinder in this case.

In each of the two cases just cited the relative motion between the
solenoid and the condenser with the radial connection is just the same,

Vol. X-l
No. 6. J

UNIPOLAR INDUCTION AND ELECTRON THEORY.

593

% •WtffffftNr.

consequently the diflferent results prove that the generation of an elec-
tromotive force in a conductor is not simply a question of the relative
motion of the conductor and the solenoid which furnishes the magnetic
field. There is indeed no good reason for expecting the observed effect of
the electromotive force to depend simply on the relative motion of the
conductor and solenoid, for the observer with his electrometer ^nd other
apparatus is an equally important third system to be considered in
specifying the motions, and so there is no conflict with relativity theory.

Experiments.

There is still another variation of the experiment, namely, to test
whether or not the inner cylinder becomes charged when the cylindrical
condenser with radial connection is rigidly connected with the solenoid
and the whole system rotated. I have recently completed an experi-
ment begun some time ago which confirms Professor
Bamett's negative result with a stationary condenser
and rotating solenoid, and confirms and gives more
exact results on the experiment of Mr. Kennard with
both solenoid and condenser rotating. The apparatus
used was the following: a solenoid A 29 cm. inside
diameter, 60 cm. long, 55 turns per cm. of length,
mounted to rotate about a vertical axis at speeds up
to 1000 R.P.M.; a cylindrical condenser 5C of sheet
copper mounted coaxially with the solenoid, outer
cylinder B of condenser 25 cm. diam., 60 cm. length,
with closed ends, except that shielded connection to
the electrometer ran through a central hole in top end ;
inner cylinder Cio cm. diam., 33 cm. length, supported*
by hard rubber blocks; a copper strip DE, running
diametrically across the inner cylinder and out nearly
to the outer cylinder, by means of which the inner cylinder could be
connected at will with either the outer cylinder, by pushing down the rod
EFf or connected with the electrometer by pushing down the electrometer
connection DG. The electrometer used was one made for this purpose
with small quadrants and a very light silvered mica needle, sensitiveness
.87 X io~* volts per division. The capacity of the shielded wire leading
to the electrometer, which was placed across the room from the rotating
apparatus, was considerable, and so the capacity of the cylindrical con-
denser was only .125 the capacity of the whole system when the electrom-
eter was connected.

I. Experiment with Cylindrical Condenser Attached to Solenoid and

Fig. 1.

594 ^^^- ^' FBJGRAU,

Rotating with It. — ^The solenoid was kept rotating steadily at a speed
of about 900 R.P.M. With no current in the solenoid the connection
was made between the two rotating cylinders by pushing down the rod
BC to touch the strip AB, and the whole was grounded through the
central wire AD. Then in turn the current was switched on the solenoid;
the central connection AD raised; the rod BC raised to break connection
between the outer cylinder and the now insulated inner cylinder; the
current switched off the solenoid ; connection of the inner cylinder with
the electrometer made by lowering the central connection AD; and the
electrometer deflection observed. The same cycle of operations was
then performed with the solenoid current reversed, and finally as a check
the same cycle, but without any solenoid current. The results in a
set of 10 measurements varied, for the double deflection, solenoid current
direct and reversed, from 47 to 50, mean 48.6, electrometer scale divisions,
or .00424 volt for double deflection, .00212 volt for deflection from
one cycle of operations. As the capacity of the inner cylinder was only
.125 that of the whole system when connected with electrometer, the
potential to which the inner cylinder was charged by rotating in the field
of the solenoid was .0170 volt. That no appreciable deflection of the
electrometer was obtained when the cycle of connections was per-
formed with no current in the solenoid simply proved that the inner
cylinder and electrometer connections were well shielded electrostatically.
In all the experiments the outer cylinder was constantly earthed.

To determine the E.M.F. that might be expected in a conductor,
such as the strip AB, rotating at the speed used in the field of the solenoid
with the current used, a copper brush was held against the outer cylinder
near the level of the strip AB connecting the two cylinders and with
the electrometer the potential difference was measured between this
brush and the central connection AD. This was .0206 volt. Assuming
the field in the solenoid at this level to be uniform and subtracting the
E.M.F. induced in the part of the strip AB inside the inner cylinder,
there is left ^ of .0206 = .0161 as the E.M.F. in the part of the strip
between the two cylinders, as against the .017 volt measured as the
potential to which the inner cylinder was charged. Allowance for non-
uniformity of field in the solenoid would bring a still better agreement
for the two results.

2. Experiment with Cylindrical Condenser and Connections Stationary ^
Solenoid Rotating. — Confirming Bamett's result, on carrying out the
cycle of connections described above with the cylindrical condenser
stationary, the electrometer indicated no charge at all on the inner
cylinder.

Na*6!^'] UNIPOLAR INDUCTION AND ELECTRON THEORY. 595

The answer given by experiment to the question of the seat of the
electromotive force in unipolar induction is therefore that it is in the
moving conductor and that without a moving conductor there is no such
E.M.F., regardless of whether the system which produces the magnetic
field is rotating or not.

Theory of Unipolar Induction.

The same answer to the question, without need of recourse to such
open circuit experiments as described, is given by even the crudest
electron theory of conduction. For on an electron theory the current in
a stationary solenoid would be viewed as a steady circular transport of
electrons around the solenoid, and the rotation of the solenoid would
amount simply to superposing a similar steady circular transport of all
the electrons, positive and negative, in the material of the solenoid.
But a steady current in a fixed circuit certainly does not affect a
neighboring stationary charge or electron. The solenoid with its current,
whether stationary or rotating, can therefore not have any action on a
neighboring stationary electron; but the current in the solenoid, through
its magnetic field, does act on neighboring moving electrons, hence in a
unipolar circuit it must necessarily be the moving electrons, t. e., the
electrons of the moving conductor, which are acted upon by the E.M.F.,
whether the solenoid be spinning or at rest.

In the early days of electron theory Sir Joseph Larmor^ stated the
same result as an application of his theory. In the article referred to
above Professor Howe arrived at the same conclusion by reasoning
based wholly on the fact that the mechanical force on an element of
length of wire carrying a current across a magnetic field does not depend
on the motion of the magnetic field or the source of the field, but only
on the magnitude and direction of the field at the element considered.
It is to be remarked, however, that the accepted facts as to the mechanical
force acting on a conductor in a magnetic field do not of themselves
entirely justify Professor Howe's argument. For let us imagine a horse-
shoe magnet carried along with its poles either side of a long straight
wire through which a current runs. Let us assume for the sake of
simplicity that conduction in the wire is by convection of the negative
electrons only and that the magnet is moving with the same speed as these
electrons. We might claim on the one hand that the transverse force on
the wire in the field does come from the negative electrons moving (with
respect to the observer) across the magnetic field as it exists at the instant,
without regard to the motion of the magnet that produces the field. Or

1 Larmor. Royal Society Transactiona, 1 895^1 , p. 727.

596 GEO, B. PECRAli, [^SS

on the other hand, conceiving the relative motion of the electrons in the
conductor with respect to the magnet to be the cause of the force on the
conductor, we might very well claim that when the magnet is stationary
the force on the conductor comes from the force on the negative current
electrons arising from their motion with respect to the magnet, and
that when the magnet moves along as fast as the negative electrons
there is no longer any force on the negative electrons of the current, but
that there is and should be just the Scime force on the conductor, arising
from a force acting on the positive electrons of the conductor, which now
have relative to the magnet the same velocity as the n^;ative electrons
in the first case, except in the opposite direction. The real trouble with
adopting the second line of argument, which would be compatible with
the localization of the electromotive force in the stationary part of a
unipolar induction circuit, is that we should be adopting a too naive
relativity principle, which misleads by not taking account of the fact
that we are supposing the force on the wire to be that manifest to the
observer, who is not at rest with respect to either the magnet or the
negative electrons in the current. Such a relativity theory would for
example teach that the force between two electrons moving abreast
with identical velocities in parallel lines is, to a stationary observer,
just the same as if both electrons were stationary. This conclusion is
at variance with the Lorentz-Einstein relativity theory and with every
theory of the electrodynamics of moving charges. According to accepted
relativity theory two electrons stationary with respect to the observer
have only the electrostatic repulsion, but if they are moving with respect
to the observer the force between them appears to the observer to be
something different from the electrostatic force. Nothing in the experi-
ments on unipolar induction is at all at variance with the Lorentz-
Einstein relativity theory.

The Lorentz electron theory may be readily applied to a more com-
plete analysis of the unipolar problem and connected questions, and may
make clearer certain points.

The two fundamental phenomena of ■ electromagnetic induction may
be given the following expression in terms of the electron theory: —

(a) A force may be exerted on a sUUianary electron by suitable
motions or variations of magnets or currents in the vicinity; that is,
electronically interpreted, by suitable motions of electrons in the vicinity.

(b) A force acts in general on an electron moving in a magnetic field,
which force is perpendicular both to the instantaneous magnetic field
intensity and to the velocity of the electron.

It is the essence of the Lorentz theory that the phenomena (a), (&)

No^df] UNIPOLAR INDUCTION AND ELECTRON THEORY, 597

and (c, electrostatic phenomena) are assumed to be independent of
and superposable upon one another. Therefore the total force on any
electron is the vector sum of three parts: (a) the force arising from the
velocities and accelerations of neighboring electrons, which force is
independent of the motion of the electron under consideration, + (6) the
force arising from the motion of the electron under consideration in a
magnetic field, + (c) the force arising from the electrostatic action of
neighboring electrons.

By well-known mathematical development from the Maxwell field
equations in the Lorentz form, the quantitative expression for the
effect (a) of moving charges on a given electron of charge e comes out
e/c(— (dA/dt)), A being the vector potential at the momentary position
of e; for the effect (b) of the motion of e with velocity i; in a magnetic
field H it is e/c[v X H] ; for the electrostatic effect — e grad <^, <^ being
the electrostatic potential. Hence

e dA e _.

force on electron = Tr + "[^XHj — e grad 6.

c at c

Applying this to finding the force on an electron in the vicinity of a
spinning solenoid, we may at once conclude that the first term, which is
a force not dependent on the velocity of the electron under consideration,
vanishes, for, as reasoned above, the transport of electrons in a rotating
solenoid merely adds to the transport of electrons in the current when
the solenoid is stationary a similar circular transport, by the rotation,
of equal numbers of positive and negative electrons, and so the whole
effect is just that of the current in the stationary solenoid, which is nil
on a stationary electron. The second term obviously vanishes for
stationary electrons, and we may also suppose the third term, referring
to the static field, to vanish. Hence, there is no force on a stationary
electron, therefore no E.M.F. in stationary conductors in the vicinity
of a steadily spinning solenoid carrying a constant current. On the
other hand, since the second term does not vanish when the electron is
moving, there is an E.M.F. on electrons in moving conductors, which
is easily seen to be quantitatively just what would be computed on the
"rate of cutting magnetic lines" scheme, supposing the lines of the
magnetic field to remain stationary with the conductors rotating.

Although the conclusion that no electromotive force is set up in sta-
tionary parts of the circuit in the unipolar induction experiment follows
so inmiediately from electron theory, many well-trained physicists and
engineers at first are inclined to disagree with the conclusion. They
are accustomed to the experience that in general the motion of the

59^ GEO. B, PEGRAM.

source of a magnetic field sets up an electromotive force in neighboring
conductors, and they have not examined the rate of variation of the vector
potential, which is the only function adequate to express the electro-
motive force at a point in a stationary conductor. It is not sufficient to
know the magnetic field intensity at the point and its variation in time
and space. The analysis must be carried back to an expression of the
effect at the given point of each neighboring moving element of charge
or electron, that is the vector potential

A — — vector 2) — ,
4xc r

the vector summation being for all the moving charge (summation for
all moving electrons), r the distance from the point for which the vector
potential is calculated to the position where the element of charge was at
time r/c earlier, and u the velocity of the element of charge at that time.

Since the vector potential at a point is a function of the positions and
velocities of all the neighboring electrons, the reason why there is no
rate of change of the vector potential in the neighborhood of the rotating
solenoid or magnet used in a unipolar induction machine is that, statis-
tically considered, the configuration of positions and velocities of the
electrons of the solenoid or magnet remains constant. As electrons of
the solenoid or magnet move out one side of a stationary element of
volume as many more move with the same velocity into the same element
of volume.

An example may be cited of a case in which the magnetic field intensity
and its time and space variations are known at a point, yet from these
nothing can be said as to the electromotive force in a conductor (or the
force on an electron) at that point. Imagine two long solenoids one
inside the other with axes parallel and currents through them so their
magnetic fields just neutralize each other inside the inner one. Now
suppose a conductor inside the inner solenoid and suppose this solenoid
is moved a little transversely. Where the conductor is the magnetic
field intensity is constantly zero, and its time variation and its space
variation are therefore also zero, hence these give us no indication of
any probability of a force on the electrons of the conductor as the inner
solenoid is being displaced transversely. Consideration of the rate of
variation of vector potential in this case determines at once that it is
not zero as the solenoid is moved transversely and that the E.M.F. in
the conductor is just the same as though the outer solenoid, which
neutralizes the magnetic field, were not present. Of course we may
adhere to the "cutting of line of force" computation of the electromotive
force if we say that we must treat the fields of the two solenoids as entirely

No'dfl UNIPOLAR INDUCTION AND ELECTRON THEORY. 599

separate and distinct in their effects and say that when the solenoid is
moved transversely all its lines of force move with it, even though the
experimental facts of unipolar induction preclude our saying that the
lines move with the solenoid when it rotates. If, however, we once
b^n this analysis of a magnetic field at a point into discrete constituents,
we should logically continue it down to the magnetic fields of the indi-
vidual electrons, which amounts to just the same thing as the vector
potential analysis.

A still more familiar case in which we have an induced E.M.F. in a
r^on where the magnetic field is constantly zero is that of a point near
a transformer, say with a toroidal core and closely wound primary
through which an alternating current flows. The moving electrons in
the primary coil and the core give a varying vector potential at points
in the surrounding space, although there is never any magnetic field
there. The usual explanation on the "cutting lines of force" basis is
to say the lines of force spring out and in, but if they do so, and yet
have at no moment a density different from zero at points outside the
core and winding, where there is no magnetic field, they must be springing
in and out with infinite velocity; which makes an unsatisfactory repre-
sentation.

One more case of unipolar induction may be referred to. Suppose an
insulated copper wire runs through a hole along the axis of a cylindrical
bar magnet and out through a radial hole to a collector ring near the
middle of the magnet. If the magnet be set in rotation and a stationary
loop of wire have its ends brought in contact with the axial end of the
copper wire and the collector ring respectively a current will flow around
the circuit. Neglecting for the sake of the argument the small magnetic
field in the axial and radial parts of the hole in which the copper wire
lies, we may say that in this arrangement there is no conductor moving
across a magnetic field, so the induction of E.M.F. is not to be explained
as in the unipolar induction cases already discussed. But here the vector
potential is varying at the position momentarily occupied by an electron
in the radial copper wire, in a manner quite analogous to the variation
at a point near a solenoid in transverse motion, and so again the seat
of the E.M.F. is in the moving wire, although it is now to be referred
to the first term in the Lorentz expression for force on an electron, instead
of to the second. Of course the vector potential at all points in the
stationary part of this circuit is in general varying on account of the
asymmetry of configuration of electrons and velocities resulting from the
radial hole in the magnet, but this variation integrates out for a whole
turn of the magnet. The shift in this example from the second term of

600 GEO, B, PEGRAM, [^»

the Lorentz expression for force to the first term as the cause of the uni-
polar or acyclic electromotive force is suggestive of the close relation
between the two terms, of, in fact, the relative nature of the two. An
observer stationed on a transversely moving solenoid observing an
E.M.F. in a ''stationary" but to him apparently moving conductor,
would attribute the electromotive force to motion of the conductor in
a magnetic field, the second term in the Lorentz expression, while a
stationary observer seeing the solenoid move would refer the separation
of the charges in the stationary conductor to the variation of the vector
potential with the motion of the solenoid; which is to say what is now
generally accepted, that the quantities involved in all electromagnetic
induction are the positions and motions of the electrons relative to the
observer.

Summary. — Experiments confirm the results of Barnett and of Kennard
showing that in unipolar induction the "seat of the electromotive force"
is in a moving conductor and is entirely independent of the rotation of
the magnetic field.

The facts of unipolar induction are in accord with the theory of
relativity. The theory of unipolar induction emphasizes the importance
of electron theory and the vector potential function in the discussion of
such questions.

lIS^^'] METALLIC CALCIUM. 60I

THE SPECIFIC RESISTANCE AND THERMO-ELECTRIC

POWER OF METALLIC CALCIUM.

By C. L. Swisher.

THE electrical properties of metallic calcium have received little
attention. In 1857 Matthiessen^ determined the conductivity.
He records two determinations at room temperature. Again in 1905 the
specific resistance was determined by Moissan and Chavanne* at room
temperature only. The thermo-electric power has not previously been
studied.

The calcium used in the present work was obtained from Kahlbaum.
Chemical analysis showed a purity of 99.57 per cent.

One of the chief difficulties in working with metallic calcium is due to
its great chemical activity. A fresh surface of calcium exposed to
ordinary air soon becomes covered with a whitish coating which has a
very high electrical resistance. The depth of this coating increases with
the time of exposure. The presence of moisture in the air or a higher
temperature greatly increases the activity.

A large number of liquids were tried with the hope of finding one in
which the metal could be stored until ready for use. The reaction in
paraffin oil was small, but appreciable. The best results were obtained
by using benzol which had been in contact with calcium carbide for some
time before the metallic calcium was introduced. This liquid was not
entirely free from action with the calcium, but the activity was small.

The chemical reactions were especially troublesome in making measure-
ments. In measuring resistance, for example, a coating of oxide would
change the effective diameter of the wire and also render the contacts
unreliable. For these reasons an atmosphere of ordinary air could not
be used. Hydrogen and nitrogen atmospheres were tried in turn with
the result that each formed a troublesome compound with calcium at
temperatures above 300 degrees Centigrade. Resort to work in a vacuum
seemed to be the only practical solution of the difficulty. The method
of securing and maintaining the vacuum and also suitable contacts is
shown below.

» Phil. Mag. (4), Vol. 13, P, 81, 1857.

* Comptes Rendus. Vol. 140, P, 124, 1905.

602 C. L. SWISHER. [

The metal as it was received was in more or less cylindrical masses
three to four cm. in diameter and six to eight cm. in length. A good bit
of trouble was experienced in getting the material into the form of a
wire suitable for measurements of resistance. The method finally em-
ployed was to saw the large cylinders lengthwise into pencils three to
four millimeters square and round them off with a knife or a file. They
were then drawn, under oil, until the length was approximately doubled.
Drawing beyond this amount did not prove successful, as the wire became
brittle. The wires as used were about .25 cm. in diameter and about
15 cm. in length. The resistance of such a wire of calcium is of the
order of .001 ohm. Measurements could be made to three significant

figures.

Method of Measurements.

The difficulty of securing low resistance leads and reliable contacts
which could be used inside a furnace led to the abandonment of the use
of the Kelvin double bridge. The potentiometer method was sub-
stituted. (See diagram. Fig. i.) In this method the resistance of leads

-^WWWWV |i|rjiji|i|i :

"i^^lwW 0—0 -^^^^^

^ • ©

Fig. 1.

and contacts need not be negligible,, and are not necessarily constant.
A resistance in the leads only makes the apparatus less sensitive, and
for the small changes of resistance which occurred this was not serious.

Apparatus.

The potentiometer used was a Leeds and Northrup instrument, as was
also the galvanometer. The ammeter was a milHvoltmeter and shunt
carefully calibrated against a Weston laboratory standard.

The furnace was made from a i^ in. gas pipe. This was permanently
connected at one end to a motor-driven vacuum pump which was kept
running during measurements. The pipe was wrapped with asbestos
paper and wound uniformly with nichrome ribbon for a length of about
two feet. This was in turn covered thoroughly with asbestos. This
arrangement gave a very uniform temperature over the length of the
specimen. The connections to the specimen were made by means of
four 3/16 inch iron rods arranged as indicated in the diagram (Fig. 2).

Vot. X.!
Na6. J

METALLIC CALCIUM,

603

The iron rods had holes near the ends to allow the specimen to extend
through, and were filed down so as to make sharp contacts The speci-
men was held in place by sharpened screws in the ends of the rods.
These rods were insulated from each other and from the furnace by means

zzs

ZSEZ2SE:

A /^irnffSfiisL

S^0c^merf

ItmrntO'Jvncftcfh

j^f^^rfar /.mm.

#fhy

V2Z2S

rrtmuL

Fig. 2.

of porcelain tubes surrounding them. At the open end of the furnace
the leads passed through a rubber stopper which fit airtight into the end
of the furnace and had the advantage of being readily removable

This arrangement did not remove all the air from the specimen, but
the pressure was well below one mm. of mercury and for the time (about
two hours) of a run very little oxidation took place. Repeated checks
showed that, for the potentiometer method, the slight oxidation which
occurred was insignificant.

The above arrangement made it possible to measure the resistance at

temperatures ranging from room temperature up to about 600 degrees

Centigrade.

Data and Results.

The following tables give the values obtained for the first heat on each
of three specimens. The curve. Fig. 3, shows the three sets of data

Fig. 3.

Specific resistance of caldum, and temperature. Ordinates « p X xo*. Abscissae « degrees C.

6o4

C. L. SWISHER.

plotted as a single curve. In the tables:

R = resistance of specimen in ohms.
B = temperature in degrees Centigrade.
L = length of specimen.
D = diameter of specimen,
p = specific resistance of calcium.
a = temperature coefficient of resistance.

specimen O.

^Xio».

#Xio«.

/?xio».
2.59

#Xxo».

1.15

24"

4.6

384"

10.36

1.30

83"

5.2

2.82

480"

11.28

1.61

164"

6.44

3.02

524"

12.08

1.92

244"

7.68

3.20

568"

12.80

2.11

296"

8.44

3.44

608"

13.76

L — 10.4 cm., D — .23 cm., a — .00365.

Specimen P,

.828

22"

4.76

1.79

374"

10.30

.986

102"

5.66

1.82

390"

10.45

1.21

170"

6.97

1.94

451"

11.12

1.40

245"

8.07

2.24

518"

12.30

1.54

292"

8.85

2.40

600"

13.80

1.73

358"

9.95

•

L — lo.o cm., D — .270 cm., a — .00377.

Specimen Q.

.604

22"

4.78

1.10

282"

8.73

.634

41"

5.02

1.14

310"

9.04

.649

48"

5.14

1.21

343"

9.58

.672

60"

5.33

1.245

358"

9.87

.702

86"

5.56

1.335

397"

10.59

.742

104"

5.88

1.42

452"

11.23

.764

111"

6.05

1.52

500"

12.05

.821

138"

6.50

1.645

563"

13.05

.925

178"

7.33

1.75

600"

13.88

1.000

211"

7.93

1.768

605"

14.00

1.052

262"

8.36

L — 7.5 cm., D « .275 cm., a ^ .00364.

The following table gives points at 50-degree intervals taken from
the curve :

No. 6. J

METALLIC CALCIUM.

605

Data from the Curve,

pXio^.

pxic^.

0**

4.27

350**

9.74

50**

5.08

400**

10.50

lOO*'

5.86

450**

11.27

150**

6.63

500**

12.05

200**

7.41

550**

12.82

250*^

8.20

600*^

13.60

300**

8.96

Summary.

Matthiessen's results give an average specific resistance of 7.7 X lO"*
ohms per c.c. at a temperature of 16.8® Centigrade. Similarly Moissan
and Chavanne give an average value of 10.5 X lO"* ohms per c.c. at 20®
Centigrade. Each of these determinations was made using a bridge
method which I found to be unreliable for continued work with calciumt
because of the variable contacts. Values taken from my curve show a
specific resistance of about 4.6 X lO"* ohms per c.c. at 20® Centrigade.
The specific resistance increases linearly, within experimental error, up to
about 13.6 X io~* ohms per c.c. at 600 degrees Centigrade. The tempera-
ture coefficient is thus constant throughout this range, and has a mean
value of .00364. In this, calcium is seen to agree very well with other

pure metals.

Thermo-Electric Power.

In measuring the thermo-electric power the potentiometer was used,
and much the same method of protecting the specimen was employed as
in resistance measurements. In this case the container was closed at
each end with a rubber stopper and was connected to the pump at the
middle.

The thermo-electric power of calcium was measured against annealed
platinum and then plotted against lead, as usual. The platinum used
was carefully checked against a piece of pure test lead.

The contact between platinum and calcium was secured by silvering
the platinum to a piece of stiff iron wire which extended out through
the rubber stopper at the end of the container. This method of securing
contact served two purposes: First, as the container was exhausted the
rubber stoppers were pressed in by the air and firm pressure between the
platinum and the calcium was thus insured. Second, the iron wires
could be twisted or turned around so as to cut fresh surfaces of contact
for each reading if desired.

A platinum-rhodium wire was fused to each platinum wire at the point
of contact with the calcium. These Pt, Pt-Rho junctions were (iarefully

6o6

C. L. SWISHER.

calibrated against the department standard and used to determine the
temperatures of the two Pt-Ca junctions. (See diagram, Fig. 4.)

Fig. 4.

The specimens for thermo-electric measurements were not drawn,
but simply cut out from the original masses. They were about 7 cm.
long and about 1.2 cm. in diameter. A small hole was bored in each
end of the calcium to receive the iron wires carrying the Pt, Pt-Rho
junctions. The calcium was heated by placing it inside a furnace which
consisted of a heavy porcelain tube wrapped with asbestos and nichrome

Heating Coif

Sfiec/meo

Fig. 5.

f^eMffTuSe

ribbon. (See diagram, Fig. 5.) The tube and calcium together were
then placed inside the container mentioned above.

Data and Results.
The following tables show the results of two to three heats on each of
three specimens.

E =8 thermo-electromotive force between Pt and Ca.

^1 = temperature of hot end.

Bi = temperature of cold end.

B = mean temperature of specimen.

P = thermo-electric power of Ca against lead.

Specimen A,

First Heat

Ey^xtP.

9]~9i.

-Px«o».

EXt^.

•t-Bx,

PXi^.

270
360
610

30.5

125**
158*'
219**

9.8

9.1

10.85

625
910

46
60.5

225**
299**

•

11.10
11.70

Na6. J

METALLIC CALCIUM.

607

Second Heat,

135

11.5

120**

10.40

627

41.5

294*

12.3

143

13.5

130*

9.2

612

40.0

293.5*

12.1

144.5

12.0

131*

10.6

783

47.0

337.5*

13.0

313

24.5

194*

10.7

774

46.5

337*

13.0

310

24.0

196*

10.8

990

57.1

389*

13.0

512

35.5

257*

11.7

986

57.0

389.5*

13.0

500*

34.5

257*

11.7

128
139
279
280

110
110
294
294
415
410
465
462
679

244

245

Specimen B. First Heat,

10.5
12.5
23.0
23.0

124*
131.2*
192.5*
193.5*

10.8

9.7

10.1

429
426
609
605

30.0
30.0
40.0
40.0

Second Heat,

10.5

26.5

25.1

35.5

44.5

69*

69.5*
128*
129*
170*
170*
192*
192.5*
249*

9.7
9.2
9.7
10.3
10.7
10.6
11.0
11.2
12.5

678
843
835
1,120
1,120
1.312
1,291
1,282

44.5

51.8

51.5

73.5

73.1

72.5

Specimen C. First Heat,

10.3
10.0
20.5
20

92.7*

93*
147*
148*

8.4

8.8

10.3

10.6

464
456
661
650

35.2
34.8
44.5
44.3

245*
247*
302*
304*

250*

300*

363.5*

405*

406*

221.4*
221.4*
287*
286*

11.6
11.5
11.9
11.8

12.6
13.0
12.9
13.5
13.5
13.4
13.3
13.3

10.6
10.7
11.7
11.5

Second Heat,

109

10.5

54*

9.8

527

158.5*

10 4

113

10.8

57.4*

9.8

527

43.5

166.5*

9.9

118

11.5

61*

9.5

515

42.5

167*

10.3

217

19.3

86*

10.2

920

64.7

227*

11.9

231

21.5

91*

9.7

Third Heat.

161

17.

93*

8.5

598

42.2

194*

12.0

167

16.5

93.7*

9.0

847

51.5

242*

13.7

158

16.5

94.7*

8.6

836

51.4

242.7*

13.6

161

15.9

94.7*

9.0

1,403

74.6

338*

15.4

393

30.8

145

11.1

1,650

394*

14.9

394

32.8

146.6*

10.4

1.652

86.5

396*

14.7

606

43.4

192

11.9

The following table gives the values of P at 50-degree intervals taken
from the curve.

6o8

C. L. SWISHER.

[

Data from the Curve,

/»Xxo».

-Pxxo».

50*

8.9

250*

11.85

100*

9.65

300*

12.57

150*

10.39

350*

13.28

200*

11.11

400*

14.00

Fig. 6.

Thenno-electric power of calcium against lead, and temperature. Ordinates — P X lo*.

Absdase *- degrees C.

The curve, Fig. 6, is plotted by using all the points from the three

specimens and then drawing a single line fitting all the points as nearly

as possible.

Summary.

The thermo-electric power of calcium was found to be positive with
respect to lead throughout the range investigated. The values range
from 8.9 microvolts per degree at 50° Centigrade to 14.0 microvolts per
degree at 400° Centigrade. The individual points vary considerably
from the straight line drawn, but each separate run follows the same
general direction, and I believe the line shown is justifiable at least as a
preliminary result. The Thomson coefficient is seen to be positive for
calcium.

Cornell University,
Ithaca, N. Y.,
June, 1917.

No'df'l TOTAL IONIZATION BY SLOW ELECTRONS. 609

TOTAL IONIZATION BY SLOW ELECTRONS.

By J. B. Johnson.

I. Before the nature of cathode rays was known, Lenard discovered
that these rays would pass through a thin aluminum window to the
outside of the discharge tube and that they made the air through which
they penetrated conductive.* Later experiments showed that the con-
ductivity was due not only to the stoppage of electric charges by the
air molecules, but to the production of new charges from the molecules
themselves,* i. e.y to ionization of the air. The first quantitative experi-
ments on this ionization were done by Durack,* who measured the
number of ions produced per centimeter per electron in air at a given
pressure, just after the rays had emerged through the aluminum window.
His results showed that the ionization is proportional to the pressure of
the air, as had been found to hold in the discharge tube itself*; and that
at a pressure of i mm. of mercury an electron made on the average .43
pair of ions per cm., when the velocity of the electrons was of the order
4 X 10* cm. per sec. Using j8 rays from radium, whose velocity he
estimated at 2.3 to 2.8 X 10*^ cm. per sec., he found the specific ioniza-
tion a to be much smaller, being but .17 for these faster rays.

These experiments were repeated under improved conditions by
Glasson* and by W. Wilson,* the former using cathode rays and the
latter the fi rays from radium B and radium C. By means of magnetic
deflection nearly homogeneous bundles of rays of known velocity were
obtained. Glasson used a range of velocities from 4.08 to 6.12 X lo*
cm. per sec., and Wilson used velocities from 1.24 to 2.90 X 10*^ cm.
per sec. For the value of a Glasson obtained 1 .5 when the velocity of
the rays was 4.8 X lo* cm. per sec. Both observers found that a is nearly
proportional to the inverse square of the velocity of the electrons, or

1 p. Lenard. Ann. d. Phys., 51, p. 225. 1894.

* P. Lenard, Ann. d. Phys.. 8, p. 149. 1902; ibid., 12, p. 449, 1903.

* J. E. Durack, Phil. Mag., 4, p. 29, 1902; ibid., 5, p. 50, 1903.

^ J. S. Townsend, Phil. Mag., i, p. 198, 1901; ibid., 3, p. 557, 1902; ibid., 5, p. 389, 1903.
J. S. Townaend and P. J. Kirby, Phil. Mag., i, p. 630. 1901.

* J. L. Glasson. Phil. Mag., 22, p. 647. 191 1.

* W. Wilson, Proc. Roy. Soc.. 85, p. 240. 1911.

6lO J, B. JOHNSON, [toSS

within the ranges used. From the above values of a and v the constant
Jfe is 345 X lo^' cm. for air at i mm. pressure. The loss of velocity
of cathode rays in passing through a solid was measured by Whiddington.^
Rays with velocities ranging from 5.31 to 8.58 X lo* cm. per sec. were
passed through an aluminum window and the loss of velocity measured
by magnetic deflection. If wo is the velocity of the incident rays and
X the thickness of the aluminum, the velocity of the emergent beam, r,
was found to be given by the expression

To* — V* = ex.

The value of c was 7.32 X 10^ cm'/sec* for aluminum. This relation is
in accord with the theory given by J. J. Thomson.'

That electrons lose velocity in going through matter has also been
shown by W. Wilson,* and the amount of this loss was calculated by
Seeliger* from measurements by Bestelmeyer.*

2. If we assume, in accordance with the theory of Thomson, that the
constant c is proportional to the density of the absorbing substance, the
results of Whiddington and of Glasson can be combined to give the total
number of pairs of ions produced per electron. Substitution of the value
of V from Glasson's equation in that of Whiddington gives

Vo* — "T = ex
or

a =

^Vo* — c'x '

The total ionization is then

n = I adx = * I

ro^-iP,*)^/ ^

^Vo* ~ c'x

The constant Vi is the velocity at which the electron ceases to produce
ions by collision. Kossel has shown that k depends only on the density

> R. Whiddington, Proc. Camb. Phil. Soc., 16, p. 321, 191 1.

s J. J. Thomson, Conduction of Electricity Through Gases, 3d ed., p. 378.

» W. Wilson, Proc. Roy. Soc., 84, p. 141, 1910.

* R. Seeliger, Verh. d. D. Phys. Ges., 13, p. 1094, 191 1.

* A. Bestelmeyer, Ann. d. Phys., 35, p. 909, 1911.

Ua^t^'] TOTAL IONIZATION BY SLOW ELECTRONS. 6ll

of the gas.* The above results indicate that the total ionization is inde-
pendent of the nature of the gas and proportional to the initial kinetic
energy of the electrons. The equations used, though resting on theoret-
ical considerations, were verified over only a limited range of velocities
and were found to hold only approximately even over this range. The
formula, therefore, cannot be expected to give more than the correct
order of magnitude of the number of ions produced by electrons having
velocities much outside of the range given above. Nevertheless, the
following values were calculated as an example, using for c' the value
5.4 X lo** for air at i mm. pressure.' The value of vi is taken as
.20 X 10* cm. per sec., corresponding to about 10 volts.

fiTSLys 2.5 X 10»« 8.000

6,500 volts 4.8 X 10* 293

1.000 volts 1.88 XIO* 45

100 volts .595 X 10> 4J

The value for P rays agrees in order of magnitude with the results of
Eve and of Geiger and Kovarik, given below; while the number of ions
at 100 volts is about twice as great as the result obtained in the present
experiment.

3. Measurements on the total ionization of fi rays have been made
by Eve,* and by Geiger and Kovarik.* Eve measured the ionization
at different distances from a source consisting of radium or radium B
and radium C, and from the absorption coefficient found that the total
ionization was 1.2 X 10* pairs of ions per electron. Geiger and Kovarik
found the ionization over the first ten centimeters of path of the fi rays
from various radioactive sources. After correcting for reflection and
determining the absorption,* the total number of ions produced by each
j8 particle was calculated. The results range from 3.3 X lo* to 17.3 X 10*
pairs of ions per j8 particle, the same order of magnitude as Eve's result.
In both of these experiments the coefficient of absorption was taken to
be constant for the whole path of the electrons, and refers to the loss
in numbers of electrons, not to the loss of velocity.

4. In measuring the ionization per unit path of electrons, Kossel also
found, indirectly, the total ionization produced by an electron in air.*

* W. Kossel, Ann. d. Phys., 37, p. 393, 1913.

* Obtained from c by the density law.

» A. S. Eve, Phil. Mag.. 32, p. 551, 191 1.

* H. Geiger and A. F. Kovarik, Phil. Mag.. 32, p. 604, 1911.

* A. F. Kovarik, Phil. Mag., 20. p. 849. 1910.

* W. Kostel. 1. c.

6l2 J. B. JOHNSON, ^S

His method is based on the following considerations. Let no electrons
start in a given direction in air at i mm. pressure, and let ao be the
fraction of the electrons that are stopped by collisions in one centimeter
of path. The number of the original electrons at any place x is then
given by

Let a be the number of collisions per centimeter which result in the
production of a pair of ions, and let fh be the total number of pairs of
ions made by the n© electrons.
Then

dfh — n\adx

and

fit = «oa I C^^dx

a
do

The average number of pairs of ions per electron is then

fh oc

From the latter of these ratios n was calculated. The assumption has
been made here that the electrons lose no velocity until they are stopped,
since both a© and a vary with the velocity.

The value of a was determined by Kossel for electrons having velocities
corresponding to a range of 200 to i ,000 volts. Electrons were projected
between two parallel condenser plates. A small field was applied be-
tween the plates, giving the electrons a parabolic path and causing them
to be absorbed on one of the plates. The length of path was calculated
from the velocity and the transverse field. One of the plates was
connected to an electrometer, and on this plate could be collected either
the original electrons and the negative ions produced, or the positive
ions. From this data was calculated the number of pairs of ions per
electron per centimeter of path. The values were reduced to the standard
pressure of i mm. of mercury, since the ionization was found to be
directly proportional to the pressure. The pressures actually used were
of the order .05 mm. of mercury.^

The absorption coefficient a© which was used by Kossel was determined
by Lenard over a large range of velocities.* He measured the decrease

> For a resume of the work on specific ionization see S. Bloch. 1. c, p. 580; also Frantx
Mayer, 1. c.

* P. Lenard. Ann. d. Phys., 12, p. 449, p. 7x4, 1905*

». 6. J

Vol. X.1
No.

TOTAL IONIZATION BY SLOW ELECTRONS,

613

in the number of electrons in a beam traversing a space containing a
gas at a low pressure. Careful corrections were made for secondary
radiation, diffusion of ions, reflection from the walls, scattering, and
other disturbing factors.

The variation of specific ionization with velocity was also measured
by Mayer for velocities up to 500 volts.^ His values are not reducible
to absolute measure except by comparison with those of Kossel. Taking
Mayer's value of a for air at the velocity given by 500 volts to be the
same as that given by Kossel, the total ionization can be calculated.
The results so obtained, together with those of Kossel, are given in Fig. i.
The two curves do not agree very well.

The value of a for lOO-volt electrons in different gases at the same

-Ml]

k «i^^

^^]

^_ -

^^^

}

» lA^-i

L..:\

!cH

s — '

Fig. 1.
Air.

pressure was found by Kossel to be proportional to the density of the
gas, or to its molecular weight. The only exception was hydrogen,
which gave a value four times greater than the density law would indicate.
Lenard found that for fast cathode rays the absorption is proportional
to the density of the medium, except for hydrogen which had twice the
absorption of other matter of the same density.* McLennan,* using
cathode rays, and Strutt,* using fi rays, both found that the ionization
in a given distance is proportional to the density of the gas and not
dependent on its chemical constitution. McLennan found hydrogen
normal, but Strutt obtained values twice as* great as the density law
implies. Other experimenters* have obtained about the same values of
the absorption coefficient as Lenard. The density law holds except

* F. Mayer, 1. c.

* P. Lenard, Ann. d. Phys.. 56, p. 255. 1895.

* J. C. McLennan, Phil. Trans. (A), 195, p. 49, 1901.

* R. J. Strutt, Phil. Trans. (A), 196, p. 507, 1901; Proc. Roy. Soc., 68, p. 126, 1901.

* A. Becker, Ann. d. Phys., 17, p. 381, 1905. J. Robinson, Ann. d. Phys., 31, p. 769, 1910.
S. Bloch, Ann. d. Phys., 38, p. 559. 1913. F. Mayer, Ann. d. Phys., 45, p. i, Z9i4«

6l4 J' ^' JOHNSON.

for electrons of very low speeds (below lOO volts). Hydrogen is ab-
normal, the more so the lower the velocity of the electrons. Since,
then, in the expression aja^ both quantities are proportional to the
density of the gas and depend on the density only, it follows that the
total ionization should be independent of the nature of the gas, and
depend only on the initial velocity of the electrons. Hydrogen is the
only exception found and, using Kossel's value of a, should give rise to
twice as many ions as are obtained in other gases.

Another experiment showing that the total ionization produced by
an electron is independent of the nature of the absorbing gas was made
by Kleeman.^ Kleeman found that for the heterogeneous electrons
emitted by gold when acted on by X-rays, the ratio of the total ionization
produced by the electrons to that of a rays is the same for all gases.
It had been found by Bragg and Kleeman,* and verified by Taylor,* that
the total number of;ions produced by an a particle is nearly independent
of the kind of gas. This would then apply also to heteix)geneous cathode
rays. It was also found by Kleeman that the fi rays from actinium and
the j9 rays from uranium gave the same ratio of the ionization produced
in a given distance in a gas to the ionization produced under the same
conditions in air.* The actinium fi rays differ considerably in velocity
from those of uranium, and it follows that within this range the ratio
of the ionization in the gas to the ionization in air is independent of
the velocity. This indicates that homogeneous rays, too, make the
same total number of ions in all gases. The velocities of the electrons
used in these experiments differ widely and the result can not be con-
sidered as conclusive, although furnishing strong evidence that the total
ionization is independent of the nature of the gas.

6. The problem may also be looked upon from the point of view of
the energy necessary to produce ions by collisions. The minimum
ionization potentials have been determined for the simple gases with
some accuracy.* This sets an upper limit to the number of ions that
can be produced by an electron, if we assume that it takes the same
amount of energy to produce each pair of ions, independent of the
velocity of the electron. There is nothing known to justify this assump-
tion, however; it may, indeed, be that one collision may produce several

> R. D. Kleeman, Proc. Roy. Soc., 84, p. 16. 1910.

* W. H. Bragg and R. D. Kleeman, Phil. Mag., 10, p. 318, 1905.

» T. S. Taylor, Phil. Mag., 18, p. 604, 1909; Am. Jour. Sci., 28. p. 357, 1909.

* R. D. Kleeman, Proc. Roy. Soc., 83, p. 530, 1910.

» P. Lenard. Ann. d. Phys., 8, p. 149, 1902. O. v. Beyer, Verh. d. D. Phjrs. Gcs., 10, p.
100, 1908. E. S. Bishop, Phys. Rev., 33, p. 325, 1911. J. Franck and G.' Hertz, Verh. d. D.
Phys. Ges., 15, p. 34, p. 939, I9i3' ^' Mayer, 1. c. F. S. Goucher, Phys. Rev., 8, p. 561,
1916. See also R. D. Kleeman, Proc. Roy. Soc., 84, p. 16, 1910.

VOL.X.

Na6

!^]

TOTAL IONIZATION BY SLOW ELECTRONS,

615

ions/ without using a corresponding multiple of the minimum ionization
energy. Partzsch* has measured the average energy used to produce a
pair of ions in a discharge tube. The values he obtained lie between
27.9 volts for nitrogen and 14.5 volts for helium, which values are con-
siderably higher than the minimum ionization potentials (except in the
case of helium). If these values also hold for the average energy lost
by an electron per ionizing collision outside of a discharge tube, then
the total ionization arising from electrons of a given speed in the different
gases should be inversely proportional to these numbers.

7. The total ionization by electrons, then, has been measured in only
two regions of the velocity range. These measurements have been
made by more or less indirect means, and have given no definite relation
between velocity and total ionization. A formula was found from
indirect data, which gives results of the right order for the higher veloci-
ties, but which fails completely to represent Kossel's results both as to
magnitude and to form of relation. On the other hand, there are several
lines of evidence pointing to the conclusion that the total ionization is
independent of the gas and depends only on the velocity of the electrons.

8. The object of the present investigation was to determine the total
ionization by a direct method. The total ionization produced by elec-
trons of velocities up to 200 volts has been measured in oxygen, nitrogen,
hydrogen, and helium. Electrons were generated by a hot platinum
wire, accelerated in a distance less than the mean free path in the gas,
and were then allowed to spend themselves in the gas in a large ionization
chamber. The number of positive ions produced as compared with the
number of electrons entering the chamber was then measured.

9. The apparatus as finally used is shown in Fig. 2. The heavy

-1I11--

10 cm

Fig. 2.

* J. J. Thomson, Rays of Positive Electricity, p. 48.

* A. Partzsch, Ann. d. Phys., 40, p. 157, 1913. J. S. Townsend, Electricity in Gases, p.
295. 1915.

6l6 /. B- JOHNSON. IISSS

copper cylinder C contains two electrodes, A and B. £ is a brass
cylinder, closed at both ends except for two small openings, and insulated
from C by small pieces of ebonite. ^4 is a brass rod insulated from C
by an amber plug and guard ring. Electrons from the hot platinum
wire a are accelerated toward a gauze-covered opening in the diaphragm
b, pass through the gauze c and opening d and are absorbed in the gas
in the cylinder B, where the ionization is measured. The axis of the
cylinder was placed parallel to the earth's resultant field to avoid
magnetic deflection of the electrons which would otherwise be appreciable.

One of the leads of the hot-wire cathode was a brass tube, which also
served as a focusing ring; the other was a brass rod inside the tube,
and the whole was mounted in a glass holder cemented to the outer
cylinder. A 6-volt storage battery furnished the heating current. A
small electron current had to be used and this was found to be steadier
without an oxide coating on the filament. A set of storage cells V
connected between a and b gave the electrons the desired velocity, and
by the same cells the electrons could be retarded by any potential D
in steps of 2 volts, in the space be. The distance ab was about 4 nmi.,
be and ed each about 2 mm. The holes a, 6, and e were about 5, 8,
and 10 mm. in diameter, respectively. The electrometer, used at a
sensitiveness of about 150 nmi. per volt, measured the drop of potential
over a high resistance R due to the ionization current (steady deflection
method). By this means any erratic behavior of the cathode could at
once be seen. An adjustable xylol and alcohol resistance was at first
used for R, but this was found to polarize slightly. An India-ink line
on paper gave perfect satisfaction.

10. Three different measurements could be made by changing the
electrometer connections: the original electron current, the positive
ions produced, or the sum of the original electrons and the negative
ions. The diagram shows the connections for measuring the sum of the
electrons and the negative ions. C is to earth, B to earth through
the shunted electrometer, and A is connected to the negative side of
the battery E, the other side of which is earthed. To measure the
positive ions, A was connected to the electrometer, B and C connected
to the positive side of £, the n^ative side being earthed. To measure
the original electron current, C was earthed and A and B both connected
to the electrometer and used as a Faraday chamber. The connections
were made through a commutator, not shown in the diagram, so that
the change from one arrangement to another could be made in one
operation and readings taken in rapid succession. For low pressures
the reading for the sum of the electrons and the negative ions was

Vol. X.1
No. 6. J

TOTAL IONIZATION BY SLOW ELECTRONS,

617

quite accurately the same as the sum of the other two readings. For
higher pressures, however, the first named quantity was usually a few
per cent, lower, probably because at the higher pressures a larger pro-
portion of the ions were formed near the hole and were driven out through
it by the field. For this reason, only the readings for the original
electrons and for the positive ions were used in the final experiments,
the other reading serving merely as a check.

In this way the ionization was measured at different velocities and with
different pressures. The results thus obtained for the four gases are
illustrated by the values for nitrogen, Fig. 3, which gives the number of
ions per electron, m at different pressures and different velocities. For
the lowest velocities used the ionization soon reaches a maximum as
the pressure increases, and then falls off slightly, while for the higher
velocities the maximum value comes at much higher pressures; for the
highest velocities the maximum is not reached with the greatest pressures
used. This is caused by the greater penetration of the fast rays. Unless
the pressure is high enough they strike the sides of the cylinder before
their energy is spent and do not produce as many ions as at higher
pressures. The slower electrons are comparatively easily absorbed as
the curves show. In taking these curves the potential E used to drive
the ions to the electrodes was 20 volts except for the lowest pressures,
where 10 or 12 volts were sufficient to insure saturation. There was no
appreciable additional ionization if the potential greatly exceeded these
values.

Since the space ab, where the electrons are accelerated, can not be a
vacuum but must contain gas at the same pressure as the ionization

fwWWw^^^^ 0 W^W»

Fig. 3.
Nitrogen.

chamber, it is to be expected that many electrons collide within this
distance and produce new electrons that enter the ionization chamber
with low velocities. This then necessitates a correction to the values

6i8

J. B. JOHNSON.

given in Fig. 3. The velocity distribution of the electrons was deter-
mined by applying an opposing potential D in the space be and measuring
the number which got through. The gauzes at b and c were smoked in
order to avoid 5-rays. Some curves obtained in this way are reproduced
in Fig. 4, where each ordinate represents the number of electrons with

Fig. 4.

Velocity distribution curves.

velocities greater than the corresponding abscissa. At the lowest pressure
the number of electrons does not fall off much until D approaches F,
when the number falls off quite abruptly and is zero at Z> ^ V. For
higher pressures, as D increases, there is at first a sharp decrease in the
deflection, then a more gradual slope, and finally a fairly sharp drop to
a positive value at D = V which remains constant as Z^ is further
increased. The positive deflection can be ascribed to positive ions made
by the electrons in the space be. These are swept into the cylinder with
the first 10 volts of the retarding field where there is the sharpest drop
in the curve. Ordinates should then be measured from the lowest part
of the curve, where Z> = F, except for values of D between o and 10
volts. The gradual slope of the central portion of the curve is due to
slow electrons, either new electrons produced near the gauze b or original
electrons that have lost part of their energy in collisions.

II. The method of procedure was then as follows: At the lowest
pressure to be used, the ratio m between the number of positive ions and
the number of electrons producing them was determined for a series of
different velocities, beginning with the lowest velocity at which any
appreciable number of ions was produced. At least four determinations
of m were made at each velocity, using the same or different electron
currents. Then distribution curves were obtained with the same veloci-
ties before the pressure was changed. This was repeated at a series of
increasing pressures, the lower velocities being gradually dropped and

nS!"6^*] total ionization by slow electrons. 619

higher ones used< In this way were obtained curves similar to those in
Fig. 3, and velocity distribution curves for corrections to be applied at
each point.

The distribution curves show the presence of velocities ranging from
that given to the electrons down to zero, the slope depending on the
pressure. The slow electrons produce ions as well as the faster ones,
and the ratio m does not give the ionization due to any one velocity.
To get the number n of the ions produced per electron at a given velocity,
successive corrections were applied to m from the distribution curves.
At a certain velocity, 10 volts for oxygen, no ionization could be detected.
At the next velocity used, 14 volts, the ionization was produced only by
the electrons having a velocity over 10 volts, the number of which was
found from the distribution curve at that pressure. The number of
ions n produced by an electron having a velocity in the range 10 to 14
volts could then be found by dividing tn by the fraction of all the electrons
which have a velocity in this range. The number n was used in correcting
the value for the next interval, 14 to 20 volts, and so on. The correction
formula takes the form

NioHio + Nufiu + N20 + n2o+ • • • + N^n^ = m^,

where the N*s denote the fraction of the electrons havng speeds in a
given range, as found from the distribution curve, and the n's are the
number of ions per electron at that range as previously determined.
The sum of these products is the ratio m, of the ions to the electrons as
observed, and from this n, was determined. This process was repeated
for the next higher pressure and so on until all the curves were corrected
as far as was thought consistent with accuracy. It is to be noted that
the first few terms in the correction formula are almost negligible, but
as the electron velocities increase in value the total correction becomes
considerable. The reason for this is that although there are many slow
electrons, their ionizing power is small. There is an upper limit beyond
which the corrections could not be carried because for fast rays such high
pressures must be employed that the distribution curves became very
unfavorable for an accurate determination of the exact number of the
high-speed electrons. With the present apparatus this limit was reached
at velocities corresponding to about 200 volts. At this velocity the
probable error is quite high, of the order of 25 per cent.

12. The results for the various velocities were plotted against pres-
sures, as shown in Figs. 5 and 6 for nitrogen and helium. From these
curves the values of n were taken in the r^ion where they are independent
of the pressure and in Figs. 7 and 8 the number of pairs of ions per electron

620

/. B. JOHNSON.

is plotted against the eneiigy in volts. The curves are practically straight
lines, represented by the equations

n = .0276(F' — 12) for nitrogen,

n = .0275(F' — 11) for oxygen,

n = .oasSCF — 11) for hydrogen,
and n = .0244( F" — 20) for helium.

too*

•

1
1

""r~

_^^

—

-eo

^^_

B0»

Fig. 5.
Nitrogen, corrected.

Fig. 6.
Helium, corrected.

where the electrons have a velocity corresponding to F" in volts. For
oxygen and hydrogen the curves point to about 11 volts as the energy
necessary to produce positive ions, in good agreement with the generally

J
t

X''

«?

../e

K> ..

Fig. 7.
Oxycen and hydrogen.

-_^

«•

•

» -^

xkl»

•

Fig. 8.
Helium and nitrogen.

accepted values.^ The nitrogen curve cuts the axis at 12 volts whereas
ionization has been found to begin in nitrogen at 7.5 volts* or 11.5
volts.' The helium curve points to 20 volts as the minimum ionization
potential, while ionization was observed at 14 volts. The helium prob-

' J. Franck and G. Hertz, 1. c.
« Ibid. F. S. Goucher, 1. c.
* F. Mayer, 1. c.

Na*6'!^'] TOTAL IONIZATION BY SLOW ELECTRONS. 621

ably contained hydrogen as an impurity. No attempt was made to
determine the minimum ionization potentials closely because this has
already been done by more sensitive methods.

The three most prominent facts shown by these curves are, that the
total ionization is proportional to the excess of the initial energy of the
electrons above the minimum ionization energy, at least for the lower
and more accurate parts of the curves; that the results for the four
gases are practically the same; and that the values are much higher than
those obtained by Kossel's method.

As to the first of these observations, it is seen that the electrons
with velocities just above the ionization limit are very inefficient ionizers.
Only a small fraction of them produce ions, the rest losing their energy
in inelastic collisions.^ Though the electrons can produce ions, for
example in hydrogen at about 1 1 volts, they do not average one ion per
electron until a velocity due to about 50 volts is attained. The average
energy used per ion, as found from the curves, is 36 volts for nitrogen
and oxygen, and 41 volts for hydrogen and helium, above the minimum
ionization energy. The average energies in volts given by Partzsch
for the discharge tube are 27.9 for nitrogen, 23.9 for oxygen, 27.8 for
hydrogen, and 14.5 for helium. The values are probably too low, how-
ever, as they are calculated on Townsend's assumption of perfecdy
inelastic collisions. Mayer calculates the ratio of the ionizing collisions
per centimeter to the number of "kinetic theory" collisions for electrons
of about 130 volts velocity. With the aid of this ratio the energy loss
per non-ionizing collision could be calculated by assuming that not much
more than the minimum ionization energy is used in producing ions.
Mayer's values, however, depend on the results of Kossel's experiments,
and reasons will be given presently for believing that Kossel's values of
a are not correct.

That the total ionization should be nearly the same for the four gases
is in accord with the results for electrons of higher velocities. The
interpretation of this must be that the less energy a gas absorbs in
ionization, the more it absorbs in non-ionizing collisions; or, the more
inelastic a molecule is to electrons, the more easily it is ionized. That
this is so in a general way is seen from the available data on minincium
ionization potentials and elasticity of collisions. The monatomic gases
are elastic, but require in general a higher velocity for ionization than
the other gases. Exceptions to this are mercury vapor on the one
hand, and hydrogen on the other. The differences are not so marked
in the gases used in this investigation. The curves do differ in slope by

' J. Franck and G. Hertz, Verb. d. D. Phys. Ges.. 15, p. 373. 1913. K. T. Kompton and
J. M. Benade. Phys. Rev., 8, p. 449, 1916.

622 y. B. JOHNSON.

a slight amount, and it may be that a gas like argon would show a higher
total ionization than the gases employed here.

13. The values of the total ionization as obtained by the direct method
differ from the results due to Kossel's method by a factor varying from
3 to 7. The discrepancy exists not only at 200 volts but continues up to
the higher velocities. The uncorrected curves for nitrogen in Fig. 3 show
velocities up to i ,000 volts, and these point to values as high as the straight
line relation in Fig. 8 indicates. This is larger than any ordinary eaqjeri-
mental error and must be due to a fault in one method or the other.
There are two causes that might make Kossel's values too low. The
first of these is the presence of slow electrons in the electron stream.
Correction was made for electrons stopped between the condenser plates
but not for the slow electrons entering with the stream. That these
may have been present in appreciable quantities at the pressures used
is shown by the distribution curves obtained in the present investigation.
The effect of this would be to make the values for a too low. The second
objection to the method is that Lenard's absorption coefiicient which
Kossel used is not applicable here. There are two absorption coefficients
that can be considered in connection with an electron stream. One is
that defined, and measured, by Lenard, which is the loss of numbers of
electrons from the beam. It is not concerned with what happens to an
electron after its course is changed. The other absorption coefficient is
that deduced theoretically by J. J. Thomson (I. c). This refers to the
loss of kinetic energy of the average individual electron. C. T. R.
Wilson* has shown for fast electrons that the direction of the path is
often changed while the electron continues to make ions. The same
undoubtedly takes place at lower velocities, and, though the electron is
lost from the beam, it is not lost for the purpose of ionization. The
absorption coefficient deduced from this point of view may be con-
siderably smaller than Lenard's absorption coefficient. A striking illus-
tration of the difference is given by hydrogen. The absorption of slow
rays in hydrogen is twelve times that predicted from the density law,*
and still hydrogen has been found to reflect these electrons with little
loss of energy. It seems probable, then, that if the correct absorption
coefficient were used Kossel's method would give considerably larger
values for the total ionization. There are no data available on this
coefficient, however. The discrepancy between the two methods may
be largely explained by these considerations.

14. There remains to be discussed the preparation of the gases and
the effects of impurities in them. The nitrogen was prepared by heating

» C. T. R. Wilson, Proc. Roy. Soc. A.. 87, p. 277. 1912.
' F. Majrer, 1. c.

vSx6^] TOTAL IONIZATION BY SLOW ELECTRONS. 623

a solution of NaNOi and NH4CI. After the flask and connecting tubes
had been well washed out, the gas was collected over water, which it
displaced in a large bottle. Before being used it was passed over P1O5
to remove the water vapor, and over hot copper and copper oxide to
remove oxygen and hydrogen. The hydrogen was obtained by diffusion
through a hot palladium tube. The gas was collected directly in a
reservoir connected with the apparatus. Oxygen was prepared by heat-
ing potassium permanganate and was collected over a KOH solution free
from other gases. The helium was purified by passing it over hot copper
oxide to remove hydrogen which was known to be present, and by passing
it through charcoal in liquid air to remove all other impurities. The
hydrogen was probably not entirely removed since ionization began at
14 volts instead of the accepted value of about 20 volts.

To see whether vapors from the sealing-wax joints and the stop-cock
grease, and also mercury vapor, had any disturbing effect on the results,
a trap was introduced near the ionization tube. After immersing the
trap in liquid air for a number of days no difference was observed in the
ionization or in the distribution curves. The liquid air was therefore
not used in the final observations. It is not to be expected that impurities
should have so disturbing an influence as, for instance, in minimum
ionization experiments, where the limit of an effect is measured, or in
experiments where surface films on the electrodes might collect disturbing

charges.

Summary.

1. The total ionization produced by electrons of velocities up to 200
volts has been determined for nitrogen, hydrogen, oxygen, and helium
by a direct method.

2. The total ionization in these gases is proportional to the energy
the electrons possess above the minimum ionization energy, at least up
to 150 volts.

3. The results are practically the same in the four gases, in accord
with results at higher velocities.

4. Reasons have been pointed out showing why the values for total
ionization obtained by Kossel are too low.

In conclusion, the writer wishes to express his thanks to Professor

H. A. Bumstead, who suggested the problem and whose advice cleared

away many of the difliculties; he is also indebted to Dr. H. M. Dadourian

for many valuable suggestions during the course of the work.

Sloans Laboratory,
Yalb University,
June 18, 1917.

624 P' C. BLAKE AND WILLIAM DUANE. [^SS

THE VALUE OF "A" AS DETERMINED BY MEANS OF X-RAYS.

By F. C. Blakb and William Duanb.

T^UANE and Hunt^ have proved experimentally that the voltage V
^^ required to produce X-rays of a given frequency v is determined
by the quantum relation

Energy of electron = Flf = Af, (i)

in which t is the elementary charge and A Planck's action constant.
Conversely, the law may be stated thus: the quantum relation^ gives
the maximum frequency of the X-rays produced when we apply a con-
stant voltage V to an X-ray tube. This law holds for the general
X-radiation.

It had been assumed previously by certain scientists that the laws
of the quantum emission of radiant energy applied in some way to the
production of X-rays. Attempts had been made to prove from experi-
ments that the voltage required to produce characteristic X-rays obeyed
the law. This, however, is not in general true. Although the quantum
relation gives the order of magnitude of the energy required to produce
characteristic X-rays, the law does not hold strictly for all of the char-
acteristic lines. Dr. Webster* has shown experimentally that the voltage
required to produce many of the characteristic lines is considerably
higher than the voltage calculated from the quantum law. He has shown,
also, that the highest frequency line of a series very approximately
obeys the law.

In the above-mentioned researches the experimenters employed a
Coolidge X-ray tube, in which the electrons emitted by a hot tungsten
wire and not gas ions carried the current. A high potential storage
battery of 20,000 cells generated this current at the required constant
voltage.

The frequencies of the X-rays were measured by means of a Bragg

X-ray spectrometer, a crystal of calcite serving as the reflector. The

equation

X = 2a sin B, (2)

» Phys. Rbv.. Aug., 1915. p. 166.
* Phys. Rev., June, 1916, p. 599.

^^'] VALUE OF h BY X-RAYS. 625

in which 6 is the grazing angle of incidence and a the distance between
two successive planes of atoms in the crystal, gives the wave-length X of
the reflected X-rays.

The above quantum law, applied to the general radiation, provides a
new method, capable of considerable accuracy, of determining this
highly important constant A, or, strictly speaking, the ratio of h to e.
On this account we determined to try to improve the apparatus and to
make a series of measurements as accurately as possible.

Mr. Hunt assisted us in designing a new spectrometer. It differs
only slightly from an ordinary optical spectrometer. Two adjustable
slits in lead disks, i cm. thick, take the places of the objective glasses of
the collimator and telescope. A glass ionization chamber replaces the
telescope, and a brass tube flxed in position and provided at the far end
with a third adjustable lead slit replaces the collimator.

The brass tube extends through a brick wall into an adjoining room,
and the X-ray tube lies opposite its end : thus the spectrometer and the
X-ray tube are in adjoining rooms, and this arrangement furnishes a
very complete protection from stray X-rays.

The photograph represents the arrangement of the pieces of apparatus.
The X-ray tube lies behind the wall, the end of the brass tube through
which the X-rays come being hidden by the lead disk at the back. The
galvanized iron box surrounding the spectrometer protects the apparatus
from electrical disturbances.

The spectrometer carries two scales, one for the ionization chamber and
one for the table supporting the reflecting crystal. The settings of each
of these can be read by pairs of verniers to within about 5 seconds of arc.

The photograph shows a metal ionization chamber attached to the
spectrometer, but as we wished to use methyl iodide or ethyl bromide
as gases in the ionization chamber, and as these gases attack grease and
cement, we designed a glass ionization chamber containing neither
grease nor cement of any kind.

Fig. I represents this ionization chamber. It consists of a glass
tube AB with a very thin glass window blown in it at the end A. The
window lies toward the reflecting crystal. The glass of the window is
so thin that it absorbs very little of the X-radiation. To provide a
guard ring for the electrode, we had the steel tube C ground into a small
side tube D and another glass tube E ground into the steel tube C.
Through the tube £, and sealed in its end, passes a platinum wire, which
supports a second wire F running nearly the length of the tube AB,
A mercury jacket G surrounds the steel tube Z>, and this is electrically
connected to earth. Thus the mercury and steel tube form mercury-

626

F. C. BLAKE AND WILUAit DUANE.

[

sealed joints and at the same time an electrical guard ring for the elec-
trode. A thin piece of sheet steel, lying against the inside surface of
the glass tube AB and connected through H to a battery, acts as the
second electrode.

After filling the chamber with the methyl iodide or ethyl bromide, as
the case might be, we sealed off the glass intake tube and thus left no

— -^ f* *0f»fy

Fig. 1.

chance for the gas to escape. With this arrangement the gas in the
chamber touches nothing but metal and glass.

A fine wire above the ionization chamber passing through brass tubes
filled with paraffin connects the electrode to a quadrant electrometer
(see photograph). This rests on a fixed shelf immediately over the
crystal table of the spectrometer.

The voltage applied to the X-ray tube we measure by means of an
electrostatic voltmeter enclosed in a second large metal lined box and
joined in parallel to the electrodes of the X-ray tube.

This voltmeter has been much improved by Dr. Webster and Dr.
Clark. It consists of four large metal balls, two of them stationary and
two suspended by a bifilar suspension. The deflection of the movable
balls can be read by means of a telescope, mirror and scale. The magni-
fication of the telescope is such that i/ioth of a millimeter on the scale
can be estimated easily. The deflection of the instrument for the voltage
used amounts to about 80 cm., so that an observer keeping his eye on
the instrument, and varying a water resistance in series with the X-ray
tube, can keep the voltage applied to the tube very constant during the
experiment, in spite of a small decrease in the electromotive force of the
battery that usually occurs.

F. C. BLAKE AND WILLIAM DUANE.

Vot.X.1
Nad. J

VALUE OF k BY X-RAYS,

627

We calibrate the voltmeter during each series of measurements by
means of a current flowing from the high potential storage battery
through a resistance of 894,700 ohms. We measure this current by
means of a milliammeter which we calibrated with two entirely different
potentiometers and standard cells.

Unfortunately the zero of the instrument does not remain quite fixed.
The shift, amounting to not more than 2 mm., occurs during the first
two minutes after the voltage is applied to the instrument. In taking
readings for the calibrations we allow a short time to elapse to correct
for this shift. On account of the shift the accuracy of our measurements
is somewhat uncertain, but we would be much surprised if the error of a
single measurement of the voltage amounts to as much as i/io of one
per cent.

In setting up the electrometer we had no difficulty in placing the crystal
so that its front face lay very close to the axis of rotation of the crystal
table. In doing this we employed optical methods, using a fluorescent
screen to locate the X-ray beam. By these methods also, we determined
roughly the zero positions of the ionization chamber and of the crystal
table. Since, however, we make measurements on both sides of the

' .^.A-

Fig. 2.

zero position we do not have to determine the zeros with great accuracy,
provided that certain corrections to be described later are carefully
estimated and applied.

As Duane and Hunt pointed out in finding the value of h either the
voltage can be kept constant and the crystal table turned around (the
ionization chamber being shifted at twice the rate), or the crystal table

628

F. C. BLAKE AND WILUAJd DUANE.

and ionization chamber may be kept fixed and the voltage varied. In
either case the ionization current is measured. We adopted both
methods of procedure. The curves (Figs. 3 and 4) represent the ioniza-
tion current as a function of either the voltage or the crystal table setting.
The values of the angles and the voltages corresponding to the points
where the ionization currents vanish are the values to be used in calculat-
ing h (or the ratio of h to e) from equations (i) and (2). The distances
of the horizontal portions of the curves above the zero axis of current
indicate the sizes of the natural leak in the ionization chamber plus the
current due to stray X-rays.

Near the point where the ionization current vanishes the curves are
rounded off toward the horizontal axis. The rounding off is due to the
finite widths of the slits and of the source of rays. A glance at Fig. 2
will make this clear. F represents the focal spot on the target of the

X^MiMstimm OtwrmMt €m. Mr^trmry Units

Fig. 3.

X-ray tube. 5i, 8% and 8% represent the three slits. Evidently all the
rays striking the crystal do not make quite the same angle with the
reflecting planes, so that the reflected beam contains waves of slightly
different wave-lengths.

We adopted two methods of estimating the points at which the ioniza-
tion currents vanish. First we continued the curves downward, as
represented by the dotted lines in Figs. 3 and 4, and assumed that the
points at which the curves met the zero lines correspond to the wave-
lengths of the rays passing through the centers of the slits; and secondly,
we estimated from the shapes of the rounded-off portions of the curves

Na6. J

VALUE OP h BY X-RAYS.

629

the points at which the ionization current actually did vanish. Evidently
the latter points correspond to the extreme X-rays that just graze the
edges of the slits. The differences between these two values should
represent one half the angle between the extreme rays as represented in
Fig.. 2. As a matter of fact a very close agreement between the results
of the two methods of estimating the vanishing point proves that this
way of correcting for the finite widths of the slits and source is sub-
stantially correct.
Evidently the angular breadth of the beam of X-rays will depend upon

«7

«f-

.-J— — I Lj— .

nw nm Mjr

Fig. 4.

the relative magnitudes of the source and of the slits and upon their
relative positions. Fig. 2 represents the three cases that occurred in
our experiments.

In the arrangement marked A the slits S% and 8% determine the angular
breadth of the beam. If we call h one half the angular breadth of the
beam, then h represents the correction to be added to the grazing angle
of incidence. The value of h may be calculated at once. It appears

630

p. C. BLAKE AND WILUAit DUANE.

rSscoicD
LSbsiks.

from the geometry of the figure that

St X

and that

from which

tan 5 = —
2y

tan 6

St + St

In the experinients the

2{x + y) •

X + y being the distance between the slits,
values of a, b and c were about

a = 26 cm., 6 = 40 cm., c = 56 cm.

In one experiment the dimensions of the slits and source had values
such that the breadth of the focal spot on the target and the width of
St determined the angular width of the beam of X-rays. B of Fig. 2
represents this case.

In a third experiment we arranged the apparatus so that Si and St

controlled the width of the beam, as
shown at C of Fig. 2. A method of
calculating the slit correction d simi-
lar to that explained above applied
in this case.

In measuring the widths of the slits
we proceeded as follows. We removed
the crystal table and placed the slits
in line with each other. We then
gradually closed the slit the width of
which was to be estimated, and meas-
ured the ionization current for differ-
ent positions of the slit's micrometer
screw. Fig. 5 represents the actual
measurements taken in two of the de-
terminations. The numbers on the
horizontal axis represent the amounts in hundredths of a millimeter by
which the slit was closed from the width that it had during the experiment.
The ionization current vanished at a certain point, and evidently the dis-
tance from the vertical axis to this point gives the original effective width
of the slit. Owing to a slight lack of alignment of the lead sides of the
slits the ionization current generally vanished before the sides came in
close mechanical contact with each other.

Fig. 5.

/••

VOI.X.

Na6

VALVE OP k BY X-RAYS.

631

It may be noted here that the slit correction does not usually amount
to as much as 2 per cent, of the grazing angle of incidence, which gives
an idea of the accuracy with which the critical limiting wave-length
can be estimated from the curves of Frgs. 3 and 4.

In order to find out whether in cases A and B the angle measured to
the slit in front of the ionization chamber was twice the grazing angle
of incidence, we made several tests as follows. We set the crjrstal table
at a given angle and then by moving the ionization chamber step by step,
measuring the ionization current in each position, we sought out the
position of maximum intensity. We then turned the crystal table
through an angle of (180^ — 2 ^) and again sought out the position of the
maximum intensity.

Fig. 6 represents two pairs of such maximum curves. The first pair

k- M 1 i^^ i- 'U [\

«$!• *fr IJtf* t4S*

/••

^fmf €,

Fig. 6.

corresponds to the arrangement represented by A in Fig. 2, and the
second by B in Fig. 2. In the latter case the center of gravity line
of the curve does not appear to be quite vertical. The fact that the form
of the curve does not reverse for reflection on the other side of the zero
line proves that the source itself causes the lack of symmetry. On
account of this lack of symmetry we take the position of the center of
gravity line at the bottom to represent the true reading of the instrument
for the center of the beam of reflected X-rays.

It appeared early in our work that the angle between the two center
of gravity lines was not exactly twice the angle between the reflecting

632 F. C. BLAKE AND WILUAM DUANE.

I^anes of the crystal in its two positions. In other words, the ionization
chamber angle was not always twice the grazing angle of incidence.
Sometimes it was greater and sometimes less than the grazing an^ of
incidence. This discrepancy is due largely, if not entirely, to the fact
that the X-rays are not all reflected from the surface of the crystal.
Some of these penetrate a considerable distance into the crystal and are
reflected at planes below the crystal surface.

The penetration of rays into the crystal causes a certain error in cal-
culating the values of the X-ray wave-lengths, unless it is corrected ior.
These corrections must be applied to all measurements made with the
apparatus arranged as represented by A and B in Fig. 2. No such
correction is required, however, in the arrangement represented by C
in Fig. 2; for in this case the ionization current does not depend upon
the exact position of the ionization chamber, as long as slit 8% is broad
enough to alldw the whole reflecting beam of X-rays to enter the ioniza-
tion chamber.

The method we adopted for making a correction for the penetration

Fig. 7.

of rays into the crystal may be explained as follows: Let -4, in I., Fig. 7,
represent the slit through which the X-rays come from the X-ray tube,
and let B represent the slit in front of the ionization chamber. Let O
be the axis of rotation of the spectrometer, and J{, the radius of the circle
on which the slits move. Suppose that the spectrometer has been set
up so that the center of the beam of X-rays passes through the axis of
rotation 0, and further, suppose that the slit B has been placed at the
center of the reflected beam of X-rays determined as described above.
Call EE' the effective reflecting plane of the crystal. The effective
reflecting plane of the crystal we define to be the plane in the crystal
such that, if all the X-rays penetrated to it and were there reflected
to the ionization chamber, the effect in the ionization chamber would
be the same as the sum of the effects due to the various reflections

flS^i^'] VALUE OF h BY X-RAYS. 633

actually taking place at the successive planes of atoms in the crystal.
In order to obtain the correction to be made, if this effective reflecting
plane does not coincide with the axis of rotation we must determine how
far the effective reflecting plane lies from the axis of rotation. This
may be done by means of the data obtained from the above-mentioned
maximum intensity experiments as follows: In I., Fig. 7, the grazing
angle of incidence is the angle ACE^ which we call $\ This angle is
given by the readings of the crystal table verniers. The ionization
chamber verniers give us the angle FOB, which we call a'. If 7' repre-
sents the small angle at B, it appears from the triangle OBC that
2 d' = a' + 7'. Calling OC b we have

sin 7' b

sin 2$' R '

and if d is the perpendicular distance from the axis of rotation to the
effective reflecting plane,

i? sin 7'

d = 6 sin d' =

2 cos ^

/ •

This gives us d in terms of known quantities. Now suppose' that the
spectrometer has been set up for making the measurements of the value
of A, and that the ionization chamber angle is exactly twice the angle
made by the crystal planes with the zero line. II. in Fig. 7 represents
the arrangement. Under these conditions, if the effective plane ££'
does not lie on the axis of rotation 0, the X-ray that enters the ionization
chamber will not be the X-ray that passes through 0, but the one that is
reflected at D at the foot of the perpendicular line from 0 on the effective
plane ££'; for this is the ray such that the angle of incidence equals the
angle of reflection.

Evidently the grazing angle of incidence ADE' (called now %") from
which the wave-length is calculated is not the same as the angle between
the effective reflecting plane and the zero line.

Call a" the ionization chamber angle FOB. This by the setting up
of the instrument is twice the crystal table angle FOH. Then, calling
^' the small angle at 5, we have the relations %" = J^" -f- /3', and

d sin ff

^"'M;-'')

From the previous experiments we know the value of d, and from this
we get the value of j8', which must be added to \^a" to give us the
true grazing angle of incidence.

634

p. C. BLAKB AND WILLIAM DUANB.

•»q»!»M

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I
i

iS^^] VALUE OF h BY X-RAYS. 635

If the angles are small, we may take

j8' = (?' - 3^'

and to get the grazing angle of incidence we must add /3' to the observed
angle between the reflecting planes and the zero line.

In practice we measured this angle by taking the readings on both sides
of the zero line so as not to have to determine the zero with great accuracy.

In some of the observations we did not have the observed angle
between the two positions of the reflecting planes in the crystal quite
equal to one half the angle between the two positions of the ionization
chamber. This introduces a second correction, which may be calculated
in a manner similar to that for the last correction. Calling the observed
crystal table angle 0 and the observed ionization chamber angle a, this
correction amounts to

and must be subtracted from 6 in order to give the true grazing angle
of incidence.

We have therefore three corrections to make; (a) the corrections for
the slit width, (b) the correction for the fact that the axis of rotation does
not lie in the effective reflecting plane of the crystal and (c) the correction
for the fact that the ionization chamber angle may not be exactly twice
the crystal table angle. In the column of the table marked *' 6 corrected "
all three of these corrections have been introduced, where possible.
In the first two sets of observations we did not examine the relative
values of 6' and a', but the large width of the slit Sz minimized the error
thus introduced.

The curves in Figs. 3 and 4 represent the actual readings taken. We
made six sets of observations. Fig. 3 contains plots of the first three
sets. Curves III. were taken with fairly narrow slits and show that
the slopes of the curves for extrapolating back to the horizontal axis are
not straight lines, but are slightly concaved upward. We did not take
(enough points for Curve I. to determine its true shape, so we drew
straight lines down to the horizontal axis.

Fig. 4 gives the observations corresponding to the lower three lines of
the table. For each of these experiments we kept the crystal table and
ionization chamber stationary and changed the voltage applied to the
X-ray tube. In the experiment represented by II. slits St and Sz were
each .4 mm. broad. The X-ray tube, however, had been turned so as
to give nearly the line source, and the focal spot on the target proved to
be too small to fill up the dihedral angle formed by slits St and Sz. Fig.
25, therefore, represents the method of slit correction. At best we could

636 F, C, BLAKE AND WILUAJd DUANE.

estimate only roughly the width of the source, so that we reversed the
calculations with respect to this set of observations, taking the value of
h obtained by the extrapolation method as eicactly equal to that deter-
mined by the slit correction method, and by means of the geometry of
case B calculating the width of the source. This turned out to be
0.37 mm. In the table for this set of observations certain figures are
enclosed in brackets. The corresponding value of h has not been
counted in calculating the average.

Curve III., Fig. 4, represents a series of readings by the method used
by Duaiie and Hunt in which slits Si and St were very narrow, and slit
Sz wide enough to include the entire reflected beam. Within reasonable
limits this method should be independent of the position of the effective
reflecting plane in the crystal with respect to the axis of rotation. The
experiments confirm this statement.

Curves II. and III., Fig. 4, represent experiments in which the front
face of the crystal lay at exactly the same distance from the axis of rota-
tion. Careful measurements by a cathetometer shows that the front
face of the crystal was 0.275 mm. in front of the axis of rotation. Apply-
ing the formula

^ i?sin7' 280 sin (i' 46")

^ = — - ssz — ^ — - — — ssz 0.072 nun.

2 cos d' 2 cos (3° 10' 17") '

shows that for Curve II. of Fig. 4 the effective plane (for 6 ^ B') was
0.275-0.072 = 0.203 mm. below the crystal's surface.

Although the crystal face was .275 in front of the axis of rotation,
nevertheless, this method of two narrow slits between the source and
crystal gives the correct value of h as the last line of the table clearly
indicates.

Co'umns 18 and 19 of the table contain the values of A as estimated by
the extrapolation method and by the slit correction method respectively.
Column 20 gives the relative weight we have given to the different sets
of observations.

The crystal employed was calcite and the face chosen the (100) plane.

In the formula

Ve Vek 2Ve , . ^
h = — s= - - = — d sin $
V c c

we used for d the value given by Gorton^ and by Compton,* namely,
3.028 X lo*^ cm., for e MilUkan's value, 4.774 X lO"^® electrostatic
units and for c 2.999 X 10^® cm. per sec.
As the table indicates, the extrapolation method and the slit correction

» Phys. Rev., Feb., 1916, p. 203.
> Phys. Rbv.. June. 1916, p. 646.

h =
c

n2I"6^*] value of h BY X-RAYS. 637

method agree with each other very closely. Further the values of h
obtained in the various experiments differ from each other very little,
so that we now think that any error in the measured value of h must be
sought for in errors in the value of the elementary charge e and in the
value of the grating constant d. Our average for h is 6.555 X lO"*'.
This agrees very closely with the value of h calculated by Bohr's formula

^ — aT-

for Rydberg's constant, namely h = 6.544 X lO"*', the value of N
being taken as 3.290 X 10** and the value of e/m as 1.77 X 10^ e.m.u.

Using our value of h to calculate e/m we get e/m = 1.761 X 10^ e.m.u.
and using Millikan's value of e this gives m = 9.038 X io~" grams and
taking m^ of hydrogen to be 1.650 X lO"** grams we get m^/m = 1826.

Calculating a from the equation

fe^^«/48Tay^

using Millikan's value^ for k we get a = 7.612 X io~^*, taking c =
2.999 X 10^®. This in turn gives for the value of Cj, the radiation con-
stant, 1.433 cm. deg., in very close agreement with Coblentz's recent
value,* 1.432, while Warburg and Miiller* give 1.430. The quantity
<r = ac/4. comes out 5.707 X io~*.

In addition to those deduced from radiation measurements, in which
Planck's radiation formula is used, other values of this constant h have
been obtained recently as follows (using the above value of e) :

By Means of X-rays.
Duane and Hunt* 6.51 X 10-«»

HuU» 6.59 X 10-«»

Webster« 6.53 X 10-«»

Webster and Clark' 6.53 X lO"*'

By the Photoelectric Effect.
MUlikan.* 6.57 X lO"*'

Kadesch and Hennings' 6.43 X 10"*'

Sabine>« (6.58H5.71) X 10-«»

Jefferson Physical Laboratory.
Harvard University.

^ Millikan. Proc. Nat. Ac. Sci., April, 191 7.

> Coblentz, Bulletin Bur. of Standards. 12, p. 579.

'Warburg and Miller. Ann. d. Phys., 48, 410. 1915.

* Phys. Rev.. Aug., 1915. p. 166.
» Phys. Rev.. Jan., 1916. p. 156.

• Phys. Rev., June. 1916, p. 599.

' Proc. Nat. Ac. Sci., 3, 191 7. p. 181.

• Phys. Rev., 191 6. p. 379.

* Phys. Rev.. Sept., 1916. p. 221.
"•Phys. Rev.. Mar., 1917, p. 210.

638 it. C. WORTHING,

THE THERMAL EXPANSION OF TUNGSTEN AT INCAN-
DESCENT TEMPERATURES.

Bt a. G. Worthing.

Introduction.

TN a paper by Langmuir^ on the characteristics of tungsten filaments as
^ functions of temperature, there was given an equation representing
the thermal expansion of tungsten. This was quite at variance with
what the writer expected from certain results in his possession as well
as with certain results previously obtained at room temperatures by
Fink* at the Research Laboratory at Schenectady. For the region
20** C. to 100** C. Fink obtained as the thermal expansion coefficient
3.36 X 10"* i/deg., while Langmuir's results lead to very closely 75
per cent, of this value. Because of this disagreement and of the writer's
need of these results in other work, the thermal expansion of timgsten
was investigated. Following the presentation of the results to the Phys-
ical Society,* the writer has been informed by Langmuir that an error
has been found in his work and that his corrected results agree with
the writer's.

In order that more nearly correct values may appear in print, and
with the hope that some of the details of procedure may give confidence
and interest in them, the results and some of the details are here presented.

Method.

For the measurements at high temperatures, hair-pin and rectangular
shaped filaments of considerable length with fine marks scratched on
them, were mounted in long tubular glass bulbs, which were later
evacuated. It was possible to obtain lengths of approximately 18 cm.
which were satisfactorily free, at all the incandescent temperatures used,
from the cooling of the leads to the filament, and on which therefore
expansion measurements could be made. In this work a traveling
micrometer microscope and a position microscope were used. The posi-
tion microscope was always sighted on some small marker connected
with the scratch near the top of the length being measured, the traveling

> Phts. Rev., 7, p. 329, 1916; Gen. Elec. Rev., 19. p. 21X, 1916.

> Trans. Am. Electrochem. Soc., 17, p. 233, 1910.
* Jour. Franklin Inst., 181, p. 857, 1916.

NoI"6^*] THERMAL EXPANSION OP TUNGSTEN. 639

micrometer microscope on some convenient marker at the other end.
As the filament expanded or contracted with a change in temperature,
the lamp was raised or lowered so that the position microscope was
always sighted at the same marker, the error in the scales being thereby
entirely limited to that one associated with the screw of the traveling
micrometer microscope, a high-grade Soci6t6 Genevoise instrument.
Repeated measurements on the same filament with slight shifts in the
position of this instrument together with the general concordance of the
results showed that for this work the irregularities of the screw were not
appreciable. The pitch of the screw has been assumed to be correct.
Considerable care was taken, to protect the measuring instruments and
the framework holding them in position, from heating appreciably.
Any effects of this kind were within experimental limits. Temperatures
were measured with an optical pyrometer of the Holbom-Kurlbaum
type. With this arrangement the expansions from room temperature
to various incandescent temperatures were determined. In this part
the most serious difficulty arose from slight unequal shiftings of the
filament on heating which made it impossible to retain good focusing
conditions throughout a set.

For the measurements at temperatures below incandescence, a filament
somewhat over a meter in length was stretched horizontally along the
axis of a brass tube which was wrapped with insulated copper wire for
heating purposes. The filament was kept taut by means of a thread
passing over a pulley and attached to a weight. Preliminary results by
Dr. Dodge, of the University of Iowa, on Young's Modulus for tungsten
showed this to be a justifiable procedure. Two slits in the brass tube
about 10 cm. from each end permitted the making of observations without
serious end coolings. A small amount of magnesium smoke helped
greatly in making the wire visible and in affording points on which posi-
tion settings could be made. The average of many determinations with
a calibrated thermocouple taken at equal intervals along the axis of the
tube was taken as the temperature to be ascribed to any particular
condition. The same general precautions were taken here as at the
higher temperatures. In this part the most serious difficulty resulted
from the variations in temperature along the filament.

Results.

For a temperature scale based on 1336° K. as the gold point, a Cj for
Wien's equation of 14350 m X deg.,^ and the emissive powers of tungsten'

> See Hyde. Gen. Elec. Rev. ao, p. 819, 1917. The preliminary results previously referred
to were based on the same temperature for the gold point but with a Cs of 14460 /jl X deg.
• Phys. Rev.. II., 10. p. 377, 1917.

640

A. C. WORTHING.

reported elsewhere, the values for relative expanaon on heating from
300^ K. that have been obtained are shown in the accompanying figure.
Results at incandescent temperatures are given for two filaments only,
since only in these two cases was sufficient care taken in going from room
temperature to the lowest incandescent temperature. Results pn-eviously
obtained on several other filaments showed equally good agreement at
incandescent temperatures. Though for these there were relatively
large variations in going through the lower interval, their average for
this interval differed but slightly from that given on the plat. Only
one filament as shown was used for the low range of temperatures.

The experimental values are very well represented by the empirical
equation

L-Lo

4.44 X io-«(r - 300) + 4-5 X io-"(r - 300)«

+ 2.20 X io-"(r - 30o)»,

^i¥

i»IL

.010

.oof

Air

where Lo and L respectively refer to the filament lengths at 300** K.
and at the temperature T expressed in ** K. The average deviation of
the observed relative elongations from those to be computed from the
ciu^e is very closely H of ^ P^r cent. The coefficients of expansion at
300** K., 1300® K. and 2300® K. are seen to be respectively 4.44 X io~*

i/deg., 5.i9Xio-*i/d^.and 7.26X
io~*i/d^. It is to be noted that,
for the range measured by Fink,
the results here presented give a
value quite closely J^ greater than
his. It is interesting to note, in
accord with the general relation
existing between the melting points
and the coefficients of expanaon of
metallic elements for a given tem-
perature range such as between o**
C. and 100® C, that tungsten, with
the possible exception of molybde-
num, has the lowest known coeffi-
cient of expansion of all. The only
measurements which the writer has
seen relating to molybdenum are
those by Fink.* Since his results for tungsten and molybdenum fit
in with the general relations mentioned, and since the discrepancy

.4 0im

oof

Oo:l

«oo

3«» 7«o

/ie«

AM* J70«

T/i, -K

Fig. 1.

Thermal expansion of tungsten as a function

of temperature.

» Loc. dt.

Na*6^] THERMAL EXPANSION OP TUNGSTEN. 64 1

arises only on comparing his value for the latter substance with the

writer's for the former, the exception may not be real. Further, of

all elements for the region o® C. to 100® C, only carbon in diamond form

is recorded in accepted tables of physical constants as possessing a lower

coefficient of expansion than tungsten, viz., 1.32 X I0"*i/deg.

Nbla Rbsbakch Laboratory,

National Lamp Works of Gbnbral Electric Company,
Nbla Park, Clevbland, Ohio,
July, 1917.

642 W. C. BAKER, [

Sbcokd

A SIMPLE CONSTRUCTION FOR A CONDENSATION PUMP.

By W. C. Baker.

THE construction of the mercury condensation pump described
below is so simple that it may be followed even by those of the
most limited proficiency in the art of glass-blowing.

A bit of fine quill tubing is joined to the end of a " i^ inch" test tube
as shown in Fig. i, a. A small enlargement is blown in this about a
centimeter from the test tube, and the quill is drawn off to a blunt point
a centimeter farther out. The end of the test tube is then cut off so as
to leave 5 or 6 mm. of parallel wall. The piece thus made will be referred
to as the dome.

Next two '*i^ inch** test tubes are selected such that one will slide
inside the other leaving not more than 2 nmi. difference between the
inner diameter of one and the outer diameter of the other. The smaller
of these is drawn down and cut as shown in Fig. i, c. This piece will
be spoken of as the chimney.

Two bits of iron wire, of diameter about i mm. are next twisted to-
gether as shown in Fig. i, b. The outer vertical pieces should rest
snugly against the inside of the chimney and the hooks should bear on
its upper edge, while the central rod is to be cut so as to hold the dome
with its lower edge about 5 mm. below the level of the top of the chimney
(see Fig. 2).

The chimney is now slid to the bottom of the larger test tube and a
point marked on the outer tube about 15 mm. above the line where the
base of the chimney rests. Here a side tube is attached and below it is
made a local enlargement, as shown in profile in Fig. i, (f. A section on
the dotted line of Fig. i , d, is given in Fig. i , e. This passage is necessary
to allow the condensed mercury to flow back freely under the base of
the chimney.

The dome is now fitted with a distance piece to hold it central in the
tube. This is made of iron wire and is shown at/ in both figures. It is
tied on with finer iron wire. The enlargement in the quill of the dome is
to prevent the distance piece from sliding off during the adjustment of the
condenser. The assembled dome and chimney are lowered to place in
the bottom of the larger test tube and the top is drawn down and sealed
to a bit of wide quill tube, as shown in Fig. 2.

N^^'l SIMPLE CONSTRUCTION FOR CONDENSATION PUMP. 643

The condenser is made from a length of "two-inch" tube, corks being
used at both ends. Sealing wax does well for all the joints of these corks
except the lower inner one, where the heat from the condensing mercury
would soften the wax. It was found best to leave this cork rather loose

Fig. 1.

Fig. 2.

on the inner tube and then to calk the joint with thin stri[>s of ordinary
electrical insulation tape pushed in with the thin blade of a small pen-
knife. If this lower joint does not prove to be quite tight, a little mercury
may be put into the condenser to a depth of about i cm. This is sufKcient
to prevent any leakage of the cooling water.

Tubes for the entrance and exit of water are as shown in Fig, 2.

Next, a tee (A) is attached to the upper quill tube and a support put
down on top of the dome, to prevent it rising during the action of the
pump. This piece, shown in Fig. i, g, is made by drawing down to a
thin solid rod a bit of thick walled quill tube.

Mercury is used sufficient to cover the base of the chimney to a depth
of 3 or 4 mm.

A hole is cut in the middle of a square of stout asbestos board so as to
allow the pump to project below to the level of the bottom of the chimney
and the bare flame of a small Meeker or Bunsen burner is allowed to
play directly on the glass (after warming up of course). Care must be

644 ^- ^- BAKER.

taken that the pump does not project so that the flame touches glass not
covered with mercury.

These pumps are easily made and work very well with a Fleuss pump
for the fore-vacuum.

PHTncAL Laboratory.
QUKRNS UNivsRsmr,

Kingston, Ont., July 15, 191 7.

NoI"6?'] X-RAY SPECTRUM OP GALLIUM. 645

THE K SERIES OF THE X-RAY SPECTRUM OF GALLIUM.

By H. S. Uhlbr and C. D. Cooksey.

1. Introduction. — ^The tables of wave-lengths of high frequency spectra
contain no data for gallium, doubtless because this element is rare and
usually very difficult to obtain. Since we had a sufficiently large amount
of the metal at our disposal it seemed desirable to investigate its char-
acteristic radiations and thus supply the missing data. As the experi-
mental work progressed, the difficulties and sources of error inherent in
the usual method became so prominent as to cause the senior author to
make an analytical study of the general problem of determining glancing
angles, and both of us to subject the old method and a new one to very
thorough practical tests.

2. Apparatus and Adjustments. — ^The spectrograph consisted of a
Hilger, type No. 2, spectrometer remodeled to meet the special require-
ments of the problem. The collimator was replaced by two slits,
the one nearer the X-ray bulb being rigidly fastened to the original
apparatus, while the slit nearer the crystal could be slid along ways
and thus placed at different distances from the fixed slit. The distance
between the centers of the slits was usually 10.5 cm., and that between
the axis of rotation of the crystal table and the more remote slit was
13.9 cm. The jaws of both slits were made of lead 2.3 mm. thick,
their opposing edges were carefully lapped plane and parallel, and they
were so mounted as to open symmetrically and remain parallel to the
axis of rotation. The frame diaphragmed the length of the fixed slit
down to 3.25 mm. The original prism table was replaced by a triaxial
crystal holder transferred from a goniometer. The telescope was super-
seded by a pair of parallel steel guides lying in a plane perpendicular to
the fundamental axis. The rack for the plate-holder could be slid along
this track, thus enabling the observer to vary at will the distance from
the photographic plate to the chief axis. The vernier and tangent-screw
associated with the telescope arm greatly facilitated the adjustment of
the track parallel to the collimating axis, that is, to the line passing
through the centers of the slits and intersecting the axis of rotation at right
angles. In other words, the plate could be translated along the collimat-
ing axis. The plate-holder was removable, its incidence or front face

646 H, S. UHLER AND C. D. COOKSBY. [iSSS

was covered with black paper (of the kind in which dry plates ordinarily-
come wrapped), and it accommodated plates 2 in. wide and 10 in. long.

The X-ray bulbs were very skilfully made by Mr. A. Greiner, vice-
president of the Green and Bauer Company, Hartford, Conn. The
anode target was water-cooled and its copper-tungsten surface was
covered with a thin sheet of nickel to which the pure gallium, when
liquefied in warm water, readily adhered. The tube was clamped in
such a position as to cause the anticathode surface (which was inclined
at 45® to the long axis of the tube) to be vertical and neariy edge-on
to the collimating axis.

A very important item in the final assembling of the bulb consisted
in the thin aluminium window (of thickness 0.012 mm.), through which
the primary radiations passed with but slightly diminished intensity.
To enable this foil to withstand the excess in pressure of the atmosphere
over the low pressure inside the bulb, it was waxed over a small slot
cut in a brass collar which covered the end of a lateral tube having the
following approximate dimensions: length 5.5 cm., diameter 3 cm.
The vertical and horizontal edges of the slot measured 3 mm. and 0.5
mm., re3pectively.

The pressure within the bulb was maintained at the best value by
means of two mercury diffusion pumps in tandem. These pumps and
their accessories were designed, made, and loaned to us by Professor
B. B. Boltwood. The bulb was excited by an "Ideal Interrupterless"
X-ray current generator purchased from the Kny-Scheerer Co. The
alternative spark gap was usually set at 4 in., occasionally at 3 or 5 in.
The current through the bulb averaged 5 milliamperes. Especial care
was taken to line up the spectrograph both in altitude and azimuth so
as to cause the collimating axis to coincide with the line passing through
the center of the aluminium window and that of the focal spot. All
final adjustments were based on photographic data.

3. Methods and Measurements. — (i) The "old" method consisted in
keeping the plate-holder at a constant distance from the axis of rotation
of the crystal while taking exposures both on the right and on the left
of the direct or central image. This length was so chosen as to make the
distance from the axis of rotation to the latent image equal to that from
the axis to the center of the fixed slit. This slit was usually so narrow
(0.02 mm.) that small, but arbitrarily made, changes in the position of
the plate-holder seemed to exert an inappreciable influence on the width
of the photographic lines.

In order to subject the old method to as fair a test as possible special
attention was given to the determination of the length of the normal

Vai.X.1
Na6. J

X-RAY SPECTRUM OF GALLIUM.

647

dropped from the axis of rotation 0 to the plane of the gelatin MN. A
rectangular steel template ABCD had one edge AD lapped plane so
as to make good contact with the gelatin side of the developed plate.
(The frame of the paper screen was removable.) This edge was longer
than the distance between the extreme right and left spectral lines in
the iSrst order. (The plates were always clamped flat both in the plate-
holder and on the comparator, since the glass usually had very noticeable
curvature.) A fine fiducial line L was scratched on a piece of white
celluloid which was inlaid flush with the upper surface of the steel at the
side opposite to the straight edge. The reference line and edge were
parallel, and their constant distance apart was measured on the same
comparator as the spectrograms. To this distance must be added the
length of the perpendicular^ between the fiducial line and the axis of
rotation. This changeable length was measured by the aid of an
auxiliary comparator having a travelling
microscope with parallel lines in the focal
plane of the eye-piece. The pitch of the
screw of the latter comparator was cali-
brated in terms of that of the larger
one first mentioned. The scale reading
corresponding to the axis of rotation
was found by making successive settings
on a second fiducial line L' when in the
two possible positions (i, i') parallel to
the reference line on the steel template,
that is, when the spectrometer table was
turned through 180®. The fiducial line
near the axis was marked [in a bit of
celluloid which was mounted on a special tripod replacing the crystal
holder. To avoid gross errors several such lines were scratched on the
white surface and different lines were used in successive measurements
of the same length. All of the linear quantities could be determined to
an unnecessarily high degree of accuracy (o.ooi mm.).

The crystals were adjusted, by the aid of a compound microscope,
so that their front surfaces coincided as nearly as possible with the axis
of rotation. These adjustments did not, of course, entirely eliminate the
two fatal errors inherent in the present method : (a) the mean "reflecting "
plane of the space grating does not coincide exactly with the outer surface
of the crystal, and (6) the photographic plate does not return to precisely
the same distance from the axis when removed from the plate-holder
(for development or distance tests) and then returned to the same.

Fig. 1.

648 H. S. UHLER AND C. D. COOKSEY. ^SSS?

(ii) The new method consisted in taking two exposures (right and
left) with the plate near the crystal and two more with the plate remote
from the grating, the slits having equal widths. An accurate steel parallel
with a suitable back-stop left no doubt concerning how far the plate had
been translated along the collimating axis. By taking a short exposure
for the lower half of the central image before putting the crystal tripod
in place — the plate-holder being in one extreme position — ^and by making
a like exposure for the upper half of the direct image after the character-
istic radiations had been impressed and the crystal table removed — the
plate-holder now occupying the other extreme position — a criterion for
the adjustment of the plate-holder guides was obtained at once. A com-
parison of the distance apart of the two images of a given spectral line
on the left with the homologous distance on the right indicated whether
the plane of the plate-holder was normal to the direction of translation.
The small errors in adjustment and construction of the apparatus were
subjected to computation and found to be negligible, since the corrections
never exceeded 0.2".

It may not be inappropriate, at this juncture, to lay emphasis on some
of the points of advantage of this method: (a) it is independent of the
position of the mean "reflecting" plane of the crystal with respect to
the axis of rotation, (6) it involves no uncertainty as to the amount of
displacement of the plate, since the plate is constrained to move the
same distance as its holder, (c) the numerator of the ratio for the tangent
of twice the glancing angle is equal to the linear displacement of the
spectral image of the same wave-length on the same side of the central
image, hence by measuring the distance between homologous parts of
the two images (on the same side) errors due to asymmetry in the dis-
tribution of radiation over the breadth of the image as well as to the
depth of the silver grains in the gelatin are minimized, {d) the steel
parallel or etalon can be measured on the same comparator as the plate,
thus avoiding relative calibration of the pitches of different screws, (e)
errors of adjustment and construction of the spectrograph can be readily
determined and the corresponding corrections easily applied, (/) within
certain limits, it does not matter where the back-stop is placed, in other
words, the interval of displacement of the plate-holder may be at any
reasonable but unknown distance from the crystal, and (g) within the
same limits, the "focusing" is independent of the wave-length. It may
also be added that we found the new method to be much easier and less
time-consuming than the fixed-plate process. The chief disadvantage
lies in the fact that the displacement of a given line (3.3 ± cm.) is much
less than the distance (12.0 d= cm.) between the right and left images in

VOL.X.1

Na6. J

X-RAY SPECTRUM OF GALLIUM.

649

the "old" method. This limitation can be removed by making the
apparatus large enough. Since the exposure times for the comparatively
soft radiations of gallium were usually 5 or 7 minutes, and never exceeded
15 minutes for the weak Kfii line, the method should not require exces-
sively long exposures for any of the radiations which do not necessitate
the use of a vacuum spectrograph for their investigation.

4. Experimental Residts. — ^The numerical data are given in full in
Table I. The upper and lower sections of the table refer respectively
to calcite and rock salt. The second, fifth, and sixth columns taken
together indicate that the experimental conditions were varied as much

Table I.

Plate
No.

Bxpoaure
Date.

Width

of
Fixed

Slit.

Mm.

Width

ofMov.

able

Slit.

Mm.

Distance
from

Plate to
Axis.
Mm.

Distance
Plate was
Trans-
lated.
Mm.

•

^1-

-^01

May 29

0.04

125.71

•

12« 47'

21,"

12*"

45' 19,"

" 30

3.3

11°

28'

16,"

" 30

II 11

3."

44' 494"

14,"

June 28

0.02

125.43

II II

23,"

45' lie"

'* 28

II 1

32,"

II 19^//

42,"

«. 29

II II

25,"

" 11,"

37,"

" 29

II II

15,"

44' 59,"

29,"

.. 29

29,"

«i 29

123.38

II II

19,"

45' 2,"

31,"

July 2

0.02

70.079

II II

11."

" 3,"

" 3

II tt

13,0"

" 3io"

'* 4

X- 10

II II

1."

44' 59,"

n 9

119.55

II II

7 "
'10

45' 3io"

** 9

41,"

" 10

28io"

June 16

119.65

13° 47'

22,"

13°

44' 57,"

" 16

3.3

II II

34/'

45' 12,"

" 26

123.13

II II

27,"

" 3,"

" 26

12°

22'

42"

1. 27

II II

26,"

" 0,"

23"

as possible. Two calcite crystals and one rock salt crystal were used,
and each one was removed from the holder and readjusted at least
twice. The (100) cleavage faces were used in all cases. The former
material gave perfect definition, but the latter produced slight irregu-
larities in the images. Since the problem which we had set for ourselves
was to determine the glancing angles with respect to calcite, we con-
sidered the very accurate determination of the ratio of the grating space
of calcite to that of rock salt to be an entirely independent question.
In other words, the rock salt was employed because a sufficiently satis-

650 H. S, UHLER AND C. D, COOKSEY. [sbrbs.

factory reduction factor, if present in the literature of the subject, has
escaped our notice, and it was desirable to obtain wave-lengths on the
same basis as the tables of Si^bahn and others.

The symbols 7«,i 7«i. and y^^ denote the glancing angles of the at,
au and fii lines of the K series of gallium, in the order named. The
relative intensities of the au ot%, and ft lines are roughly proportional to
5, 4, and 2, respectively. The subscripts in the last three columns are
the weights assigned to the associated angles in forming the general
mean values. The weights for the rock salt data are quite independent
of the indices of relative importance for calcite. The Kfit line must be
very weak for gallium since we were unable to differentiate it (with
short and long exposures) from the continuous background or "white
radiation." This weakness of the Kfit line seems to be fairly general
since the corresponding wave-length is lacking for 24 irr^^larly dis-
tributed elements in Siegbahn's table of the K series. In this connection
it may not be superfluous to call attention to the fact that with long
exposures, narrow slits, and faint lines it is absolutely necessary to rotate
the crystal, for we found it quite easy to arbitrarily produce spurious
fine lines from the white radiation by keeping the space grating in a
fixed position.

The data in Table I. lead to the following weighted mean values for
the glancing angles of gallium :

7.. = 12** 47' 15" ± 2"

Calcite

7«, = 12^ 45' 5" ± 2"

it: ^

Rock salt

7fl, = Il'*28'30"ii=2

r 7.. = 13' 47' 28
- 7.. = 13^ 45' 4"

7p, = 12^ 22' 32"

Assuming the grating space of rock salt to be 2.814 X io~* cm. (E.
Wagner) and weighting the preceding data we find the grating space of
calcite to be 3.0307 X io~* cm.^ From the 15 values of the wave-length
of each of the a lines, and from the 1 1 determinations of that of the ft
line, the unweighted mean wave-lengths are found to be

^ai = ( 1. 341 61 ii= 0.00004) X 10"* cm.,
Kx = (1.33785 =t 0.00004) X io-« cm.,
Xfli = (i. 20591 ii= 0.00006) X 10-* cm.

1 While writing the present paper the July number of the Physical Review was received.
On page 95 we notice that F. C. Blake and William Duane take 3.027 X io~* cm. for calcite.
The two values differ by 0.12 per cent., which seems quite satisfactory under the circum-
stances given in the above text. The value based on Millikan's datum for e is (3.030 ±
o.ooi) X 10"* cm., with which our value agrees absolutely.

VouX.
Na6.

X-RAY SPECTRUM OP GALLIUM.

651

As we are not aware of any reason why our data should be influenced
by greater systematic errors than those given by other investigators, and
since the wave-lengths heretofore published are usually carried out to three
decimal places, but never to more than four (and even then for wave-
lengths less than one angstrom), it seems just to conclude that the
relative values of the three wave-lengths printed above are accurate to
one or two more decimal places than have been previously attained.
Be this as it may, the appreciably greater concordance of the seconds of
arc obtained for the ai and at calcite glancing angles (fix does not show
enough contrast with the background to justify comparison) by the
"method of displacement" as compared with the old method favors
the opinion that imperfections in the space gratings will constitute the
chief factor which will ultimately limit the precision of the determination
of relative wave-lengths.

By applying the method of least squares to the calculation of the
parameters of Moseley's linear law and interpolating for gallium (iV^ = 31)
the wave-lengths of the as, ai, ft, and ft lines are found to be 1.341 A.,
1.337 A., 1.205 A., and 1.191 A., respectively. The data used in the
computations were taken from one of Si^bahn's tables (based on the same
grating space for rock salt), four elements above and below gallium being
involved. As presupposed, this rare element falls in line perfectly with
the other chemical elements and the laws discovered by Moseley. The
agreement between our experimental wave-lengths and the predicted
values is much closer than was expected, for the reason that the numbers
from which the latter were calculated are decidedly irregular. The
mutual inconsistencies of the borrowed data are shown by Table II.,

Table II.

Bl.

««.

«!•

- V 41"

- 0' 43"
+ 1' 17"
+ 0' 45"

(+ 0' 20")
+ 3' 31"

- V 18"
+ 1' 16"

- 1' 31"

-3' 6"

- 0' 52"
+ 1' 20"
+ 0' 56"

(+ 0' 46")
+ 3' 51"

- 1' 56"
+ 1' 2"

- r 45"

- 3' 30"

- 0' 8"

+ 1' 20"

+ 1' 15"

(+ 0' 36")
+ 2' 52"

- 1' 59"

+ 1' 30"

- 1' 44"

which gives the differences in the glancing angles obtained by subtracting
the least square from the tabulated values. The range of elements
involved is too small and the differences are too unsystematic to admit
of the alternative inference that the linear law is at fault.

652 H. S. UHLER AND C. D. COOKSEY. [toS

Again, in the vicinity of the atomic number 31 , the published data
satisfy the equation v^i = ''r^, — i^Kmi + 0.0064, which is Kossel's
formula with a correction term added. Substitution of our data in this
relation leads to the value 11.340 A. for the wave-length of the ai line
of the L series of gallium. Interpolation with the linear law for the
square root of the frequency gives Xx«, = 1 1-353 A. The agreement
between the numbers calculated by the two independent methods may
be considered very satisfactory at the present time.

In conclusion the authors desire to express their deep appreciation of
the assistance and counsel, with respect to pumps and vacuum difficulties,
so gladly given by Professor B. B. Boltwood. It may also be mentioned
that we have completed the working drawings for a large X-ray spectro-
graph with which apparatus we hope to attack a number of important
problems depending in some cases on the highest attainable accuracy.

Summary.

1. The glancing angles, with respect to caldte, of the K lines of gallium
have been accurately determined.

2. A new method for measuring glancing angles has been devised,
tested, and found superior to the older one.

3. A preliminary value for the grating space of calcite on the basis of
2.814 X lO"* cm. for rock salt has been obtained experimentally.

Sloans Physical Laboratory,
Valb Univbrsity,
July 28, 1917.

J5g-g^] DENSITY OF HELIUM. 653

A DETERMINATION OF THE DENSITY OF HELIUM BY
MEANS OF A QUARTZ MICRO-BALANCE.

By T. S. Taylor.

Introduction.

THE present investigation was undertaken for the purpose of deter-
mining the density of helium more accurately than had been
done previously by the use of a quartz micro-balance after the type which
was first described by Steele and Grant^ and later used by Gray and
Ramsay* in their determination of the density of radium emanation.
Some time after the present investigation was begun Aston* described a
simple form of a micro-balance for comparing the densities of small
quantities of gases with considerable accuracy.

Several balances were constructed similar to the ones used by Steele
and Grant^ and by Gray and Ramsay* and it was found that these
balances having knife edge and plane supports failed to have an entirely
reliable zero position of equilibrium under similar conditions. It was
therefore decided to try a balance of the Nemst type and the one herein
described and used was found to be entirely reliable and satisfactory for
accurate comparison of the densities of gases.

Apparatus and Method.

The balance consisted essentially of two parts: a framework of small
quartz rods having a bulb and counterpoise, and a large quartz rod bent
up in the shape of a flattened U between the l^s of which the framework
was suspended by quartz fibers. A sketch of the balance is shown in
Fig. I.

The framework, which constituted the main part of the balance, was
made in the shape of a flattened rhomboid of small quartz rods about
three fourths of a millimeter in diameter. A hollow bulb H about one
centimeter in diameter was attached at one end of the longer diagonal of
this framework, and a solid mass of quartz / was attached at the

> Proc. Roy. Soc.. 1909 A. Vol. 82, p. 580.
• Proc. Roy. Soc., 1910 A, Vol. 84, p. 536.
» Proc. Roy. Soc., 1913 A, Vol. 89, p. 439.
< Loc. dt.
•Loc cit.

654

T. S. TAYLOR.

opposite end of the same diagonal as a counterpoise. Such a framework
is readily made by placing the quartz rods, bulb, and counterpoise in a
form of the desired dimensions previously cut in a flat slab of graphite
and then fusing the rods together by means of the oxy-gas flame. The
entire mass of framework including the bulb and counterpoise was
slightly under one gram. From the ends of the rod LN, which was

Fig. 1.

perpendicular to the plane of the framework at its mid point, fine quartz
fibers were drawn out and being stretched taut their ends were fused at
F and E to the legs of the flattened U made of a heavy quartz rod.
Thus the framework upon which the bulb H and the counterpoise I
were attached was supported by the quartz fibers FL and NE with its
plane of figure vertical and at right angles to the line joining F and E.

The balance was adjusted so that its center of gravity was very slightly
below the line LN. This is readily done by adding small quantities of
quartz to the ends of the rods -ST, Z or those attached to I and H, The
final adjustment is obtained by holding the desired end of a rod for a
few seconds in the oxy-gas flame, thus volatilizing a very small quantity
of quartz. Quartz rods were fused at right angles to the mid point of
the support rod and to these were attached the forked supports near
I and H, as shown in Fig. i. These supports prevented the balance
from producing too great a torsion on the supporting fibers when a
considerable difference in the buoyancy upon counterpoise and bulb
existed, and also permitted the balance to move but slightly from
what might be called the equilibrium position. The equilibrium position
is that for which the line drawn through the center of the bulb H and
the counterpoise / is horizontal.

The case in which the balance was placed was a bronze casting having
Its internal cavity in the form of a cross, the same as that of the balance
and of such size as to allow the balance to be slipped readily into it.
The case being made in this shape made it possible to use a relatively
small volume of gas. It was so constructed that it could be evacuated
or withstand considerable internal pressure and remain gas tight.

X^^!^] DENSITY OF HELIUM. 655

The balance was adjusted in the manner mentioned above, so that
when it was placed in air at a pressure of about one sixth of an atmosphere,
it was in equilibrium position. This position could be observed by
looking through a window in the case at the small tip of quartz below the
counterpoise /. This was done by means of a low-power micrometer
microscope. The reading of the microscope which corresponded to
equilibrium position was 26.00. The balance thus adjusted was cleaned
by boiling in nitric acid and washing in distilled water. It was thoroughly
dried in an oven and placed in the case. The case was then made tight
by waxing and screwing down its cover.

In order to determine the density of helium in terms of oxygen, say,
it was only necessary to measure the pressures required to keep the
balance in equilibrium position when the case contained oxygen and
helium respectively provided the temperature was the same in both
cases. Their densities are to each other inversely as the corresponding
equilibrium pressures. It was not possible to adjust the pressure of the
gas in the balance case so as to bring the pointer below / (Fig. i) to the
zero position which was 26.00, as indicated by the microscope, but the
observed pressure could be reduced to equilibrium pressure from the
sensibility and the number of scale divisions the pointer deviated from
the zero position. The density of helium was determined in terms of
both oxygen and hydrogen. By the use of these gases the accuracy of
the results obtained could be checked up, as their densities are well
known. The sensibility of the balance was determined very carefully
and it was found that a displacement of one scale division from the zero
position corresponded to a change of one two hundred and seventy-fifth
of one per cent, of the pressure required to keep the balance at the
equilibrium position. If the temperatures of the gases were not the
same when the pressures were measured, the observations could be
reduced to the same temperature, say 0° C, by using the pressure coeffi-
cients for the respective gases. This assumes that the volume of the
bulb H (Fig. i) remained constant for slight changes in temperature
and for considerable differences in pressure. This assumption is without
doubt justified since the coefficient of expansion of quartz is so very
small and since the walls of the bulb were sufficiently thick as to be un-
affected by even very great pressures.

This investigation was carried out in a constant temperature room
which had a large heat capacity and hence its temperature was not
affected by small changes. For instance, it was found that one could
go into the room, take a set of observations requiring two or three minutes,
and not change the temperature to such an extent but that it would

656 r. 5. TAYLOR. S3S!

have come to an equilibrium temperature within an hour after leaving
the room. This greatly facilitated carrying on the experiment. A
large number of obeervations of the constancy of the room temperature
showed that the temperature did not change more than two tenths of a
degree during the course of a day even when the variation in the outside
temperature was quite pronounced.

The method of introducir^ the gas into the balance case and of deter-
mining the pressure of the same can be seen by referring to Fig. 2. The
balance case was joined to the system shown in this figure at Af. The
entire system shown in Fig. 2 including the balance case was evacuated.
The charcoal bulb £ was sealed off at the constriction just above the
stopcock. When working with hydrogen and helium the liquid air was
left on the bulb N. The gas in the container D was admitted and forced
over into the balance case by means of the transfer pump B. After
sufficient gas had been admitted the mercury was raised so as to come up

Fig. 2.

in the cut-off U. The exact pressure to bring up the bulb to the equilib-
rium position was obtained by changing the reservoir H which was so
arranged as to be moved through very small distances or considerable
ones as desired thus changing the volume occupied by the gas in L.

The pressure exerted by the gas in the balance case was determined
from the observations of the level of the mercury in the vessels L, H
and F by means of a cathetometer. The vessel F together with the
capillary tube attached constituted a s[>ecial barometer having as its
mercury reservoir the mercury in L to which it was attached by means
of a flexible rubber tube as shown in the figure. This barometer could
be raised or lowered at will and thus allow the mercury in f' to be kept

^^y] DENSITY OP BEUVid. 657

at the same relative position. The pressure of the gas was then obtained
in two ways: First from the levels of the mercury in ^and L and second
from the difference in the mercury levels in L and H together with the
reading of a separate barometer. The glass vessels F, L and H were
sufficiently large so that the effect of surface tension on the level of the
mercury was entirely negligible. After having finished an experiment
with any one gas it could be pumped out of the system by means of the
transfer pump and collected in a reservoir such as X. The tubes A
and V of Fig. 2 contained PiOi,

I^PARATION AND PURIFICATION OF THE GaSES.

Helium. — ^The helium was separated by Professor Boltwood from
thorianite obtained from the Galle Province, Ceylon. It was purified
in the apparatus shown in Fig. 3. The system being carefully evacuated
the helium was introduced from a container such as D. The tube M
contained fused calcium chloride. A was made of hard glass containing
copper and copper oxide and was enclosed in an electric furnace. The
tubes L, H and F contained PjO», A charcoal trap N immersed in
liquid air was inserted as shown. The transfer pump T was used to

Fig. 3.

circulate the gas through the system. The gas when purified was
collected in the container E. By repeatedly circulating the helium
through this system the impurities would be taken out by the heated
copper, copper oxide, the drying substances and the charcoal. After
circulating the gas for some time it was all collected in the container E
and that absorbed in N was pumped out of the system through P after
heating N. After regvacuating with liquid air and charcoal joined at
P the gas in E was circulated again. By several repetitions of this
process the helium was thoroughly purified. This was observed by
noting the nature of the spectrum in the discharge tubes V.

658

T. S. TAYLOR.

Sboomo

Hydrogen. — ^The hydrogen was also purified by the apparatus shown
in Fig. 3. It was generated in the cell X between nickel electrodes by
the electrolysis of a fifteen per cent, solution of sodium hydroxide.
After passing slowly through the system it was collected in a container E.
The tube A in the electric furnace contained only copper in this case.

Oxygen, — ^This was also produced by the electrolysis of the same
solution as the hydrogen and was purified by the same method except
that the tube A contained only copper oxide and the charcoal trap N
was omitted in the system shown in Fig. 3.

Results.
A series of observations which were obtained when the balance case
contained oxygen is given in Table I. The first column contains the
values of the pressures, corrected for temperature, required to keep
the balance in equilibrium. The temperatures of the room corresponding
to the pressures given in column i are recorded in column 2. This
temperature was determined by two thermometers one graduated in
tenths of a degree and the other in fifths of a degree. Smaller fractions
of a degree could be estimated. These two thermometers were compared
with a standard Reichsanstalt thermometer and the temperature readings
corrected accordingly. Column 3 gives the reading of the pointer
below / (Fig. i) as measured by the micrometer microscope. The last

Table I.

Observed Pressure.

Temperature.

Position of Pointer.

Reduced Equilibrium
Pressure.

109.12

19.24
19.30
19.30
19.32
19.34

26.05
25.9L
26.12
26.30
25.87

101.91

109.05

101.89

109.07

101.82

109.19

101.86

109.08

101.91

column gives the pressures that would be required to keep the balance
in equilibrium position (the pointer at 26.00) if the temperature were o® C.
The reduced equilibrium pressures are calculated from the relation

^• = ^'[rT^]

100

100 +

275 J

where a is the average pressure coefficient oyer the range of temperature
and pressures respectively, and the last part in brackets is tKe^cofrectton
term to reduce the pressure for the reading in question to the equilibrium
position of pointer. The values of a for the three gases used are: oxygen

VouX.
Na6.

DENSITY OP HELIUM.

659

0.0036652, hydrogen 0.0036621 and helium 0.0036626. The factor x
in the last term is the number of divisions of the microscope reading
above or below 26.00. It is positive when the reading is greater than
26.00 and negative when less. The factor 1/275 is the sensibility as
defined above. As can be seen from the reduced values, the equilibrium
is remarkably good.

Similar sets of observations consisting of from 3 to 5 measurements
were obtained for each of the gases, oxygen, helium, and hydrogen,
and the pressures for equilibrium at 0° C. calculated in each case as
above. The results are all recorded in Table II.

Table II.

Oxygen.

Hydrogen.

HeUum.

101.91

1620.2

815.54

101.89

1620.0

815.70

101.82

1619.5

815.58

101.86

1619.7

815.83

101.91

1620.5

815.77

101.89

1620.1

814.45

101.84

1619.6

815.68

101.96

1619.5

815.65

101.86

1619.2

815.49

101.90

1619.9

815.61

101.86
101.84

1619.82 =fc 0.083

815.57
815.65

101.88
101.84

815.618 lb 0.0243

101.93

101.83

101.878 ±0.0067

The column headed oxygen represents results from four separate experi-
ments, the hydrogen column two and the helium three. After each set
of observations such as the one given in Table I. for oxygen, the gas
was removed, the vessel reevacuated and a different gas let in.

From the mean values of the reduced pressures given in Table II. the
value of the density of helium was calculated with respect to both oxygen
and hydrogen. Taking the density of oxygen = 1.42900^=0.000034
the calculated density of helium = 0.17850 db 0.000015 and taking the
density of hydrogen = 0.089873 d= 0.0000027 the calculated density of
helium = 0.17848 ± 0.000012.

By use of the relation which states that the molecular weights of
gases are to each other as their ideal densities

Ml _ ^i(^ "" ^1)

660 T. S. TAYLOIL [i

it is possible to calculate the molecular weight of helium in terms of
oxygen, it being taken as 32.00. In the relation M, d, and a refer to the
molecular weight, density, and compressibility coefficient of the gas
respectively. The values of (i — a) for the gases are as follows: Oxy-
gen 0.99903, hydrogen 1.00077 ^^^ helium i. 00000. Substituting the
values in this expression the molecular weight of helium is found to be
4.0008 db 0.0005. This value is in good agreement with that found by
Heuse,^ who gives the molecular weight of helium = 4.002. His calcula-
tion was made in the same manner from his determinations of the density
of helium by direct weighing. His value of the density of helium is
0.17856 db 0.00008, which is practically the same as the one found in

the present experiments.

Summary.

1. The density of helium has been determined by the use of a quartz
micro-balance with reference to both hydrogen and oxygen and the
values found are 0.17848 db 0.000012 and 0.17850 db 0.000015 respec-
tively.

2. The molecular weight of helium has been calculated in terms of
oxygen as 32 and found to be 4.0008 db 0.0005.

In conclusion I wish to express my indebtedness to Professor Boltwood,

who furnished the raw helium, and to Professor Bumstead, as director

of the laboratory, for the facilities for carrying on the investigation.

Sloanb Physics Laboratory,
Yale University,
June 15, I9I7-

> Ber. d. D. phys. Ges.. 15. 518. 1913.

>. 6. J

Vol. x.1
No.

X-RAY CRYSTAL ANALYSIS.

66 1

A NEW METHOD OF X-RAY CRYSTAL ANALYSIS.^

By a. W. Hull.

THE beautiful methods of crystal analysis that have been developed
by Laue and the Braggs are applicable only to individual crystals
of appreciable size, reasonably free from twinning and distortion, and
sufficiently developed to allow the determination of the direction of their
axes. For the majority of substances, especially the elementary ones,
such crystals cannot be found in nature or in ordinary technical products,
and their growth is difRcult and time-consuming.

The method described below is^a modification of the Bragg method,
and is applicable to all crystalline substances. The quantity of material
required is preferably .005 c.c, but one tenth of this amount is sufficient.
Extreme purity of material is not re-
quired, and a large admixture of (un-
combined) foreign material, twenty or
even fifty per cent., is allowable pro-
vided it is amorphous or of known
crystalline structure.

Outline of Method.

The method consists in sending a
narrow beam of monochromatic X-
rays (Fig. 2) through a disordered
mass of small crystals of the substance
to be investigated, and photograph-
ing the diffraction pattern produced.
Disorder, as regards orientation of the
small crystals, is essential. It is at-
tained by reducing the substance to
as finely divided form as practicable, placing it in a thin-walled tube of
glass or other amorphous material, and keeping it in continuous rotation
during the exposure.* If the particles are too large, or are needle-shaped

> A brief description of this method was given before the American Vhytkal Society in
October, 1916, and published in this journal for January, 191 7.

'If the powder is fine, rotation is not necessary unless great precision is desired.
With crystal grains .01 cm. in diameter, or less, the pattern generally appears quite uniform
without rotation.

Fig. 2.

662 A. W, HULL. [;

or lamellar, so that they tend to assume a definite orientation, they are
frequently stirred. In this way it is assured that the average orientation
of the little crystals during the long exposure is a random one. At any
given instant there will be a certain number of crystals whose loo planes
make the proper angle with the X-ray beam to reflect the particular
wave-length used, a certain number of others whose 1 1 1 planes are at
the angle appropriate for reflection by these planes, and so for every
possible plane that belongs to the crystal system represented. Each of
these little groups will contain the same number of little crystals, pro-
vided the distribution is truly random, and the total number of crystals
sufficiently large. This condition is very nearly realized in the case
of fine powders, and may, by sufficient rotation and stirring, always be
realized for the average orientation during the whole exposure; that is,
there will be, on the average, as many cubic centimeters of crystals re-
flecting from their lOO planes as there are cubic centimeters reflecting
from III, 210, or any other plane. This is true for every possible plane
in the crystal.

The diffraction pattern should contain, therefore, reflections from every
possible plane in the crystal, or as many of these as fall within the limits
of the photographic plate. Fig. i, Plate i, shows the pattern given by
aluminium when illuminated by a small circular beam of nearly mono-
chromatic rays from a molybdenum tube. The exposure was nine hours,
with 37 milliamperes at 30,000 volts, and crystal powder 15 cm. from the
target and 5.9 cm. from photographic plate. The faintness of the vertical
portions of the circles is due to the cylindrical form in which the powder
was mounted, causing greater absorption of rays scattered in the vertical
plane. Patterns containing many more lines are shown in Figs. 6-10, where
the diaphragm limiting the beam was a slit instead of a circular aperture,
and the pattern was received on a photographic film bent in the arc of
a circle.

The number of possible planes in any crystal system is infinite. Hence
if equal reflecting opportunity meant equal reflected energy, it would
follow that the energy reflected by each system of planes must be an
infinitesimal fraction of the primary beam, and hence could produce no
individual photographic effect. It is easily seen, however, that only those
planes whose distance apart is greater than X/2, where X is the wave-length
of the incident rays, can reflect any energy at all. Planes whose distance
apart is less than this cannot have, in any direction, except that of the
incident beam, equality of phase of the wavelets diffracted by electrons
in consecutive planes. Hence the resultant amplitude associated with
any such plane is very small, and would be identically zero for a perfect

Fig. s- Tungsten X-Ray Sprctnim.

Flc. 76. Silicon Steel,

Fig. II. Graphi

Fig. 7<i. Silicon Steel. Fig. 11. Diamond.

A. W. HULL.

1%:^'] X-RAY CRYSTAL ANALYSIS. 663

lattice and sufficiently large number of planes. The total scattered
energy is therefore divided among a finite number of planes, each of
which produces upon the photographic plate a linear image of the source
(cf. Fig. i). The total possible number of these lines depends upon
the crystal structure and the wave-length. For diamond, with the
wave-length of the Ka, doublet of molybdenum, X = 0.712, the total
number of lines is 27. All of these are present in the photograph shown
in Fig. 12. For the rhodium doublet, X = 0.617, the total number is 30;
for the tungsten doublet, X = 0.212, it is more than 100; while the iron
doublet, X = 1.93, can be reflected by only three sets of diamond planes,
the octahedral (iii), rhombic dodecahedral (no), and the trapezohedral
(311). The diffraction pattern in this case would consist, therefore, of
but three lines.

The positions of these lines, in terms of their angular deviation from
the central beam, are completely determined by the spacing of the
corresponding planes, according to the classic equation nX = 2d sin ^,
where 6 is the angle between the incident ray and the plane, hence 26
is the angular deviation, d the distance between consecutive planes,
X the wave-length of the incident rays, and n the order of the reflection.
The calculation of these positions is discussed in detail below.

The relative intensity of the lines, when corrected for temperature,
angle, and the number of cooperating planes, depends only upon the
space distribution of the electrons of which the atoms are composed.
Most of these electrons are so strongly bound to their atoms that their
positions can probably be completely specified by the positions of the
atomic nuclei and the characteristic structure of the atom. Experi-
ments are in progress to determine such a structure for some of the
simpler atoms. A few of the electrons, however, are so influenced by
the proximity of other atoms, that their position will depend much on
the crystal structure and state of combination of the substance. There
is also good reason to believe that certain electrons are really free, in
that they belong to no atom, but occupy definite spaces in the lattice,
as though they were atoms.

With elements of high atomic weight, where each atom contains a
large number of electrons, the majority of these electrons must be quite
close to the nucleus, so that the intensity of the lines will depend primarily
upon the position of the nuclei relative to their planes, and only slightly
upon the characteristic structure of the atom and the position of valence
and free electrons. With these substances, therefore, the relative inten-
sity of the lines gives direct evidence regarding the positions of the atoms,
and may be used, in the manner described by the Braggs,^ for the deter-

> X-Rays and Crystal Structure, pp. 120 ff.

664 ^- ^- BULL, [^22

mination of crystal structure. The powder photographs have an ad-
vantage, in this respect, over ionization-chamber measurements, in that
the intensities of reflection from different planes, as well as different
orders, are directly comparable, which is not true of ionization-chamber
measurements unless the crystal is very large and may be ground for
each plane.

In the case of light substances, on the other hand, the intensities depend
very much on the internal structure of the atoms, and unless this structure
is known or postulated, but little weight should be given to intensity in
determining the crystal structure. Much evidence for the structure
of these elements may be obtained, however, from the observation of
the position of a large number of lines, and this evidence will generally
be found sufficient. The examples given at the end of this paper are
all elements of low atomic weight, and the analysis given is based entirely
on the position of the lines. The photographs used for the analysis
are preliminary ones, taken with very crude experimental arrangements,
and yet in every case except one the evidence is sufficient.

The method of measuring and interpreting intensity will form the
subject of a future paper.

•

Experimental Arrangement.

The arrangement of apparatus is shown in Fig. 2. The X-ray tube
is completely enclosed in a very tightly built lead box. If a tungsten
target is to be used this box should be of )^ inch lead, with an extra J^
inch on the side facing the photographic plate. If a rhodium or molyb-
denum target is used J^ inch on the side toward the photographic plate,
and 1/16 inch for the rest of the box, is sufficient. The rays pass through
the filter F and slits Si and 52, and ldX\ upon the crjrstal substance C,
by which they are diffracted to points pu pi^ etc., on the photographic
plate P. The direct beam is stopped by a narrow lead strip H, of such
thickness that the photographic image produced by this beam is within
the range of normal exposure. For a tungsten target, the thickness of
this strip should be y^ inch; for a molybdenum target about i/ioo inch.

The X-Ray Tube.

In order to produce monochromatic rays, it is necessary to use a
target which gives a characteristic radiation of the desired wave-length,
and to run the tube at such a voltage that the radiation of this wave-
length will be both intense and capable of isolation by filtering.

The relation between general and characteristic radiation at different
voltages has been investigated, for tungsten and molybdenum, by the

Na6. J

X-RAY CRYSTAL ANALYSIS.

665

author,^ and, in more detail, for rhodium by Webster and platinum by
Webster and Clark.* The results may be summarized as follows: The
characteristic line spectra are excited only when the voltage across the
tube is equal to or greater than the value V — hvje^ where h is Planck's
constant, e the charge of an electron, and'v the frequency corresponding
to the short wave-length limit of the series to which the line belongs.

With increase of voltage above this limiting voltage, the intensity of
the lines increases rapidly, approximately proportional to the 3/2 power
of the excess of voltage above the limiting value.' The following table
will show the rate of increase for the a line of the K series of molyb-
denum, as used in the experiments described below.^

Table I.

Increase of Intensity of the Km Line of Mo with Voltage.

Kilovolts

aa.

a6.

a8.

90.

3«.

34*

9».

40.

Intensity

1.25

2.75

4.80

7.30

9.60

12.65

15.2

18.5

23.4

The rapid increase of characteristic radiation with voltage makes it
desirable to use as high voltage as possible. If the voltage is too high,
however, a part of the general radiation, whose maximum frequency is
directly proportional to the voltage,* becomes so short that it is impossible
to separate it from the characteristic by a selective filter. With a
molybdenum target the best working voltage is about 30,000 volts,
with tungsten about 100,000 volts.

Filters.

Although it is impossible to produce truly monochromatic radiation
by filtering, it is easy to obtain a spectrum containing only one Hne,
and in which the intensity of this line is more than thirty times that of
any part of the general radiation. To accomplish this, use is made of
the sudden increase in absorption of the filter at the wave-length corre-
sponding to the limit of one of its characteristic series; that is, at the
wave-length which is just short enough to excite in the filter one of its
characteristic radiations. A filter is chosen whose K series limit* lies

> Nat. Acad. Proc., 2, 368, 19x6.

« Phys. Rbv., 7, 599, 1916; Nat. Acad. Proc., 3, 185, 1917.

"Webster and Clark. Proc. Nat. Acad., 3, 185, 1917.

* The general radiation of the same wave-length as the a line is included in these values.

* See Duane and Hunt, Phys. Rev., 6, 6x9, and Hull, Phys. Rev., 7, 156.

* A complete table of wave-lengths of series lines for all elements thus far investigated is
given by Siegbahn, Ber. d. D. Phys. Gesel., xa. 300, X917.

666

A. W. BULL.

as dose as possible to tbe desired wave-length on Us short wavt-kngth side.
For example, to isolate the K lines of motybdenum whose wave-length
is .712 A., the most appropriate filter is zirconinm, the limit of whose
K series is at X = .690 A. The absorption coefficient of the filter is
then a minimum for the wave-length in question, and increases rapidly
with wave-length in both directions; on the left, toward shorter wave-
lengths, it jumps suddenly by about 8-fold; on the right it increases
more slowly, viz., as the cube of the wave-length.*

If the longest wave-length in the series, which, fortunately, in the
case of the K series, is the most intense, is chosen for the monochromatic
ray, the eight-fold increase in absorption coefficient will completely
eliminate the other lines of the series, while reducing the chosen line by
only one-half. To eliminate the general radiation is not so easy.
Webster has shown' that the intensity of the characteristic radiation
increases more rapidly with voltage than that of the neighboring general
radiation, so that the higher the voltage the more prominently the line

Fig. 3.

stands out above adjacent wave-lengths, and this is the only way in
which it can be sharply separated from longer wave-lengths. If the
voltage is too high, however, the shortest wave-length end of the gen-
eral spectrum becomes transmissible by the filter, and while its wave-
length is far removed from that of the line which is to be isolated, and
it can Itself produce no line image, yet its integral effect produces a
general blackening of the plate that obscures the lines. Sharp limitation

> Hull and Rice, Prys. Rev., 8, 326, 1916.
«L. c.

Vol. X-l
No. 6. J

X-RAY CRYSTAL ANALYSIS,

667

on the short wave-length side is obtained by the selective action of
the filter.

It is necessary, therefore, to choose filter material, filter thickness,
and voltage, to correspond to the target used. For a molybdenum target,
the filter should be zirconium, and a thickness of about 0.35 mm. of
powdered zircon is sufficient^ (see Fig. 3). The optimum voltage is
between 28,000 and 30,000 volts. For a tungsten target the filter
should be ytterbium, of a thickness of about 0.15 mm., but this has
not yet been tested. A filter of this thickness of metallic tungsten or
tantalum eliminates most of the general spectrum, but leaves the j8

eo
•to

//OArv
ftoCiCryaL

C*jrrmNo.2'OL28mml¥n/tmr'

^TTiB

Fig. 4.

doublet as well as the a doublet, which is very undesirable (cf. Figs. 4
and 5). The optimum voltage is about 100,000 volts.

The effect of filtering on the spectrum of a molybdenum target at
28,000 volts is shown in Fig. 3, which gives the intensity of the different
wave-lengths as measured with an ionization chamber, so constructed
as to eliminate, nearly, errors due to incomplete absorption.* No correc-

1 The absorption of the Si and O in zircon is negligible compared to that of the zirconium,
so that crystal zircon is as efficient as metallic zirconium.

^The ionization chamber contains two electrodes of equal length. The second electrode,
the one farther from the crystal, was connected to the electrometer, and the pressure of methyl
odide in the chamber was such that the wave-lengths in the middle of the range investigated
suffered 50 per cent, absorption in passing through the first half of the chamber. The electrom-
eter deflection is proportional to /o«"'*' (i — e'^Ot where /o is the intensity on entering the
chamber, / the length of either electrode and m the coefficient of absorption of the methyl
odide. This expression has a very flat maximum for f^^ *- }, so that for a considerable
range on either side, the readings are proportional to /o.

668 A. W. HULL,

tion has been made for coefficient of reflection of the (rock salt) crystal.

The intensities of the K lines are too great to be shown on the figure,

the a line being four times and the j9 line two and one half times the

height of the diagram. A filter of .35 nmi. of zircon reduces the intensity

of the a line from 62 to 21.4; while reducing the j9 line from 39 to 2.2.

The general radiation to the left is still quite prominent. An increase

in filter thickness from .35 mm. to .58 mm. (Curve C) reduces it but

little more than it reduces the a line, so that very little is gained by

additional filtering. The sudden increase in absorption of the zirconium

is seen at Xo = 0.690 A., which is exactly the short wave-length limit

of its K series, as extrapolated from Maimer's values of the fi\ and /3s

lines of yttrium and the fi\ line of zirconium.

The effect of a tungsten filter upon the spectrum of tungsten at 110,000

volts is shown in Figs. 4 and 5. Here the critical wave-length of the

filter is at the short wave-length edge of the whole series, so that all the

lines are present. A filter of ytterbium would eliminate all but the a

doublet. Fig. 4 gives the ionization chamber measurements, uncorrected,

of the tungsten spectrum at 110,000 volts, as reflected by a rock salt

crystal. The upper curve is the unfiltered spectrum, the lower that which

has passed through a filter of 0.15 mm. of metallic tungsten. The K

lines are much more prominent in the filtered than in the unfiltered

spectrum, but the general radiation, especially the short wave-length

end, is much too prominent, showing that the voltage is too high. In

Fig. 5 the effect of the tungsten filter (above) is compared with that of

I cm. of aluminium (below), in order to show more clearly the selective

effect of the tungsten filter. The wide middle portion of the spectrum

is unfiltered.

The Crystalline Material.

The Bragg method of X-ray crystal analysis is by far the simplest
whenever single crystals of sufficient perfection are available. If, how-
ever, perfect order of crystalline arrangement cannot be had, the next
simplest condition is perfect chaos, that Is, a random grouping of small
crystals, such that there is equi-partition of reflecting opportunity
among all the crystal planes. This has two disadvantages, viz., that
the opportunity of any one plane to reflect is very small, so that long
exposures are necessary; and the images from all planes appear on the
same plate, so that it is impossible, without calculation, to tell which
image belongs to which plane. It has the advantages, on the other hand,
of allowing a definite numerical calculation of the position and intensity
of each line, and of being free from uncertainties due to imperfection and
twinning of crystals. In the latter respect it serves as a valuable check
on the direct Bragg method.

X^^^] X-RAY CRYSTAL ANALYSIS, 669

The crystalline material is, wherever possible, procured in the form
of a fine powder of .01 cm. diameter or less. This may be accomplished
by filing, crushing, or by chemical or electro-chemical precipitation, or
by distillation. In the case of the metals like alkalies, to which none
of these methods can be applied, satisfactory results have been obtained
by squirting the metal through a die in the form of a very fine wire, which
is packed, with random folding, into a small glass tube, and kept in con-
tinuous rotation, with frequent vertical displacements, during exposure.

The method of mounting the crsytalline substance depends on the
wave-length used. If tungsten rays (X = 0.212) are used, so that the
angles of reflection, for all visible lines, are small, it is most convenient
to press the powder into a flat sheet, or between plane glass plates, and
place this sheet at right angles to the beam. In this case the correction
for the difference in absorption of the different diffracted rays is negligible.
If a molybdenum tube is used, on the other hand, diffracted rays can be
observed at angles up to 180** (cf. Fig. 10), so that the substance must
be mounted in a cylindrical tube. In this case also, the correction for
absorption is unnecessary, provided the diameter of the tube is properly
chosen and the beam of rays is wide enough to illuminate the whole tube.

The optimum thickness of crystalline material, for a given wave-length,
may be calculated approximately as follows:

Let k represent the scattering coefficient and m the absorption coefficient
of the substance for the wave-length used, and /© the intensity of the
incident rays. The intensity scattered by a thin layer dx at a distance x
below the surface will be

dR = kl^^'dx.

This radiation will suffer further absorption in passing through a thick-
ness t-x^ approximately, where / is the thickness of the sheet. Heiice
the total intensity of the scattered radiation that emerges will be

R = ] kloe-'^'dx

This will be a maximum when

^ = kloie-'^' - M/e-'^O = o

where / is the thickness of the crystalline sheet in centimeters and fi
the linear absorption coefficient.

670 A. W. HULL, [IS^

If the material is in cylindrical form, the optimum diameter is slightly
greater than the above value.

Exposure.

Very long exposures, as remarked above, are necessary if a large number
of lines is desired, and it is important to increase the speed by the use of
an intensifying screen, and by bringing the crystal as close as practicable
to the tube. With rays as absorbable as those from a molybdenum tube,
it is necessary to use films, not plates, with the intensifying screen.
Under reasonable conditions, an exposure of ten to twenty hours will
produce a general blackening of the plate well within the limit of normal
exposure. Since a greater density than this cannot increase the contrast,
nothing is to be gained by longer exposure. Further detail can be
hoped for only by using more nearly monochromatic rays, screening the
plate more perfectly from stray and secondary rays in the room, and
decreasing the ratio of amorphous to crystalline material in the specimen
under examination.

Analysis of the Photographs.

A. Cubic Crystals.

The method of deducing the crystal structure from the experimental
data is very similar to that used by the Braggs, with this difference:
In the Bragg method reflections from three or four known planes are
observed, and a structure is sought which gives the spacings and intensi-
ties observed for these planes. In the method described above a single
photograph is taken, containing reflections from a large number of
unknown planes, and a structure is sought whose whole pattern of planes,
arranged in the order of decreasing spacing and omitting none, fits the
observed pattern. In both cases the method is one of trial and error,
namely, to try one arrangement after another, beginning with the
simplest, until one is found which fits.

Calculation of Theoretical Crystal Spacings.

The process of calculating the spacings of the planes in any assumed
crystal structure is as follows: The positions of the atoms are specified
by their coordinates with respect to the crystallographic axes. For
example, a centered cubic lattice is represented by a system of atoms

whose codrdinate (jc, y, z) are

m + Ka.n + H./' + H*'^^"^'"'"'

and p assume all possible integral values, and the unit is the side of the
elementary cube. The distance from any atom Xu yu ^u to a plane whose
(Miller) indices are h, k, I is, for rectangular axes,

No!"6. ]

^^NALYSIS.

Since the family/
crystal, one of ' * -
d is the distd

Xu yu ?!•

value of d
the co5r<'
smallest
the II
assur

by syt>.
whose spaciu,^

As an example,
lattice is given in full bt.

The codrdinates of the aton^.

671

(I)

m the
that
irough
smallest
values of
erving the
d spacing of
Jh equal to i,
ite in equation
lie in planes at

*ose of the group

i
7= , etc. Since both

OLcing is regular and is
ube. If the structure is

centered cube, all parallel

*um value of d need be found.

al planes, that is, those whose

and tabulated ; and it is easy,

iat no plane has been skipped

*e table.

i the spacings of a face-centered

J.).

fn + }i, n, P + yif

where m, n, and p assume all possible integral values.

The first column gives the indices of the form, the second the smallest
value of d for that form, obtained by substituting the co5rdinates of the
atoms in equation (i), and the third the same value of d expressed as a
fraction of the lattice-constant, together with its submultiples d/2, d/3,
d/4, etc., corresponding to reflections of second, third, etc., order. The
unit is the "lattice constant," the side of the elementary face-centered

672

A. W. HULL.

Table II.

TmHIc^a* t%f WAran

Smallest Distancs

ttA^marAAn 91a ft a*

Spmcing of Planea and Submvltiples dfu.

M'^t,

100

.50

.25

.167

110

.354

.177

.118

111

1
V3

.577

.289

.192

210

.224

.112

.075

310

vio

.158

.079

.053

410

V17

.121

.061

.040

320

V13

.139

.070

.046

311

1
Vll

J02

.151

.101

411

V18

.118

.059

.039

511

V27

.192

.096

.064

711

vsi

.140

.070

.047

911

V83

.110

.055

.037

322

V17

.121

.061

•

.040

533

V43

.152

.076

.051

733

1
V67

.122

.061

.041

221

•

.167

.084

.056

331

1
V19

.115

.058

.038

551

1
V51

.140

.070

.047

553

1
V59

.130

.065

.043

321

V14

.134

.067

.045

531

V33

.168

.084

.056

731

1
V59

.130

.065

.043

751

1
V75

.115

.058

.038

753

1
V83

.110

.055

.037

tSr^^] X-RAY CRYSTAL ANALYSIS. 673

cube. The table contains all planes having values of d/n greater tanh
0.12. These values are collected in Table III., arranged in order of
decreasing d/n.

For convenience of reference, the spacings, i. e,, the distance between
consecutive parallel planes of the most important forms in the four most
conmion cubic lattices are tabulated in Table III.» together with such
submultiples, d/n^ of these spacings as come within the range of the
tables. The order is that of decreasing d/n, and the table contains all
values of d/n greater than .12.* The table also contains the number of
different sets of planes in each of the given forms. For example, the
hexahedral form (100) consists of three families of parallel planes,
parallel respectively to 100, 010, and 001.

To test whether any new crystal belongs to one of the lattices repre-
sented in Table III., it is only necessary to calculate the values of d/n
from the lines of its powder photograph, tabulate them in order, and
compare this table with Table III.

The unit of d/n in Table III. is the "lattice constant," t. e., the side
of the elementary cube whose successive translations can generate the
whole lattice. To find the spacing of any set of planes in a crystal
having one of these lattices it is only necessary to multiply the value
of d given in the table by the "lattice constant" of the given crystal.

The first three lattices in the table are the regular cubic space lattices,
in which every atom is equivalent in position to every other. In all
other possible cubic lattices the atoms must be divided into two or more
classes, whose positions in the lattice are not equivalent. The first
lattice IS the simple cubic lattice^ the unit of whose structure is a cube with
atoms at each comer. The |x>sitions of the atoms are specified by giving
to each of the codrdinates m, «, p all possible integral values within the
limits of the size of the crystal. Each atom in the lattice has as nearest
neighbors six symmetrically placed atoms, which form an octahedron
about it. This arrangement of atoms is exemplified by rock salt, except
that in rock salt the atoms are alternately sodium and chlorine. No
elementary substance with simple cubic structure has yet been found.

The second lattice is a centered cubic lattice^ whose unit is a cube with an
atom in each corner, and one at the center of the cube. It may be formed
by superimposing two simple cubic lattices in such manner that the atoms
of the one are at the centers of the cubes of the other. . The codrdinates

of the atoms are therefore given by] ,,> '.'ix^.iy where

> In order to shorten the table the simple cube spacings, which are much more numerous
than the others, have not been tabulated beyond dfn » .1766.

674

A, W. HULL.

Table III.

Indies* of
Form.

PUn«

Pamiliea

BsloDging

to Form.

Spacing of Planes, Indading Svbmultipla din.

8iinpU Cnbo.

Cantarad Cuba.

Pncf-aantarad
Cuba.

Diamond.

100

1.00

.707

111

.577

100

(n » 2) .500

.500

210

.447

211

.408

(» - 2) .354

(n - 2) .354

.354

(221
1100

.3333 )
(n - 3) .3333 i

310

.3160

311

.3014

.301

111

(n - 2) .2885

.2885

(» - 2) .2815

{n - 2) .2385

320

.2774

321

.2672

100

(» - 4) .2500

(n - 2) .2500

.2500

f410
1322

.2423)
,2423 /

r4ll

1110

.2358 \
(« = 3) .2358 i

.2358 1
(« - 3) .2358 i

331

.2292

210

(n - 2) .2234

.2234

421

,2180

332

.2132

211

(fi « 2) .2040

(fi - 2) .2040

.2040

s f430
1100

.200 >
{» - 5) .200 i

(431
1510

{!5

.19601
.1960/

1511

iin

.1923 \
(« - 3) .1923 /

.19231
(n m 3) .1923 i

.19231

(» - 3) .1923 /

(520
1432

(12
124

.18561
.1856 i

521

.1826

(» - 4) .1766

(n m 4) .1766

(n m 2) .1766

(n - 2) .17^

f 530
1433

.17141
.1714/

531

.169

rioo

1221

(n - 3) .167 \
.167/

{n - 3) 467 1
.167/

f 611
1532

.1621

310

(» ^ 2) .1580

.1580

533

.1525

311

.1507

(» - 2) .1507

(n - 2) .1507

631

.1474

111

(» m 2) .1442

(n m 4) .1442

(» - 4) .1442

N?'^. ]

X-RAY CRYSTAL ANALYSIS.

675

Il^iCjMI of
rVtttkt

FamiliM
Bslontiag
to Form.

ttl^ftdiic ot Aausi, IntlttdiHf Siibmttltlpit <//«.

8iiBpl« Cube.

Centered Cube.

Fece-eeatered
Cube.

Diamond.

fllO

(n - S) .1414 )

710

.1414 [

1 543

.1414 ^

f711
1551

.14001
.1400/

.14001
.1400 »

320

.1387

f211

(n « 3) .1360 )

552

.1360 \

1 721

.1360 J

321

(««2) .1336

.1336

730

.1312

f553
1731

.1301 1
.1301 /

.13011
.1301 f

(732
1651

.12701
.1270/

100

(« * 4) .1250

(n - 4) .1250

(n - 2) .1250

(741

.1230 1

811

.1230 •

I 554

.1230 ^

733

.1222

.1222-

"410
" 322

.1212 \
.1212 i

.1212 1
.1212/

m, Uf and p have all possible integral values. Each atom in this lattice
has eight equidistant nearest neighbors, which form a cube about it.
Elxamples of this structure are iron and sodium.

The third lattice is the face-centered cubic lattice. Its unit of structure
is a cube with an atom at each comer and one in the center of each
face. It may be formed by the superposition of four simple cubic
lattices with construction points* 0,0,0; H» /^» <>I /^» o, J^; o, 3^, 3^
respectively. The codrdinates of the atoms are therefore

w, «, p,

fn + H, n + }i, p,

m + }4, n, p + }4,

m, n + yi, p + yi,

where m, n, and p have all possible integral values. E^ch atom in this
lattice is surrounded by twelve equidistant atoms which form a regular
dodecahedron about it. Examples of this structure are aluminium,
copper, silver, gold and lead.

> The term "construction point*' is used to denote the position of some definite point,
which may be looked upon as the starting point (aufponkt) of each lattice, with respect to
the coordinate axes.

676 A. W. HULL. g

The fourth lattice is known as the diamond type of lattice, and is
exemplified by diamond and silicon. It may be formed by the super-
position of two face-centered lattices, with construction points o, o, o,
^^'^ M> /^» 3^ respectively. The coordinates of the atoms are therefore

w, n, p,

m + Hf ^ + Hf Pf
fn + }4, n, p + }4,

w, » + H. /> + H.

m + H.n + H.P + H.

m + H.n + H.P+H,

fn + H.n + H^p + H.

where tn, n, and p have all possible integral values.

•Each atom in this lattice is surrounded by four equidistant atoms,
which form a tetrahedron about it. The tetrahedra, however, are not all
similarly situatofl, half of the atoms being surrounded by positive tetra-
hedra, and the other half by n^ative tetrahedra. In this lattice suc-
cessive parallel planes are not all equidistant. In those forms whose
indices are all odd numbers, as (751), (533), the planes are arranged in
r^^ularly spaced pairs, the distance between members of a pair being
one fourth the distance between consecutive pairs. In all other forms,
that is, those whose indices are not all odd, the spacing is regular.

B. Crystals Other Than Cubic.

In the case of crystals belonging to systems other than the cubic,
the procedure is not so simple. It is necessary to make a separate calcula-
tion, not only for every kind of atomic grouping, but for every different
ratio of the axes or angle between axes. When these axes are not known
from crystallographic data, as in the case of graphite, for example, a
great many trials have to be made before the correct one is found.
Also, in the case of oblique axes the formula for the distance between
planes is less simple. However, when the crystallographic data is
reliable the process is not difficult. A few examples will be given below
for illustration and reference.

The general formula for the distance from a plane A, k, I (Miller indices)
to a parallel plane through the point Xu yu ^u referred to any system of

VOL.X.1

Na.6. J

X-RAY CRYSTAL ANALYSIS.

677

axes X, Yt Z, having angles X, m> ^ between the axes YZ^ XZ, and XY
respectively, is^

hxi + Jfeyi + fei — I

d =

h cos V cos fjL

I h cos yL

I cos V h

k I cos X

+ *

cos V k cos X

+ /

cos y I jfc

/ cosX I

•

cos ft / I

cos fjL cos X Z

I cos V COS n

•

COS V I COS X

cos/b( cosX I

(2)

For the three rectangular systems, the cubic, tetragonal and ortho-
rhombic, X, Ht and V are each 90® and equation (2) reduces to equation
(i). For the tetragonal and orthorhombic systems, however, and in all
the other systems except the cubic and trigonal, the coordinates xi, yu
Zu and the indices h, k, and / are not all measured in the same units.
The products hxi, kyi, Izi, of the numerator are of zero dimensions, but
the values of A, k, and / in the denominator contain the units, and must
be replaced by A/a, k/b, l/c, where a, ft, and c are the unit axes of the crystal
in the X, F, and Z directions respectively. This gives:

For the tetragonal system

^hxi + kyi + fei - I

where c is the axial ratio of the crystal ; and for the orthorhombic system

hxi + kyi + fei — I

d =

s/{h/ay + k'+(l/cy'

(4)

where a and c are the lengths of the shorter lateral axis and vertical
axis respectively.

For the hexagonal system, if two pf the horizontal axes, 120® apart,
are taken as X and F, and the vertical axis as Z, X and n are each 90^
and V 120®, and equation (2) reduces to

> This formula is easily obtained from the fundamental equation

rf — «i cos a + y I cos /S + ii cos y ^ p*
by substituting for cos a, cos 0, cos 7. and p their values in terms of h, k, /, X, n, and 9
given by the equations:

cos a — r + m cos r + « cos n ^ hp

cos fi ^ I' cos r + m + « cos X — *^

cos 7 — /' cosM + w» cos X + « ^ Ip

I' cos a -^m cos /S + « cos 7—1

where cos a, cos /9, cos 7 are the direction cosines, and /', m, n the direction ratios of the

peri>endicular p from the origin to the plane kkl.

678 A. W. BULL. ®SS

tei -f kyi + tei - I ..

^4/3(A* + A* + **) + (i/i?* '

where c is the "axial ratio" for the particular crystal species.

For the trigonal system, in which \ ^ n ^ v, equation 2 rMuces to

{hx\ + kyi + Izj — i) v^i + 2 cos* X — 3 cos* X

" ^(A* + ** + ?) sin«X + 2{hk + « + *0(co8*X - cosX) '

For the monoclinic system X and v al-e each 90® and e<}uation (2)
becomes

, hxi-\- kyi + tei — I , V

sin*Ai

where a and c refer to the lengths of the clinodiagonal and vertical axes
respectively, the orthodiagonal axis b being taken as imity.

Finally, for the triclinic system^ for which the general equation (2)
must be used, it should be noted that in order to use the equation for
numerical calculation, the quantities h^ k, and I in the denominator
should be divided by the corresponding axial lengths a, b^ c.

Standard tables of calculated spacings, like Table III. above^ cannot
be given for crystal systems other than the isometric, since the axial
ratios and angles are different for each crystal. By way of example,
however, the spacings of three hexagonal lattices having the axial ratio
1.624, which is the accepted value for itiaghesium, are given in Table IV.
The first is a simple lattice of triangular prisms, the length of whose side
is taken as unity and whose height is therefore 1.624. It is one of the
regular space lattices. The positions of the atoms in this lattice are
given, in hexagonal co5rdinat(es (see equation (5) above) by {x, y, z e=)m,
», pc where each of the coSrdinates f«, «, and p assumes all possible
integral values, and c is the axial ratio. The second lattice in Table IV.
is composed of two of the above triangular lattices intermeshed in such
a way that the atoms of the first are in the centers of the prisms of the
second and vice versa. It differs but very little from the so-called hexag-
onal close-packing, which is one of the two alternative arrangements which
the atoms would assume if they were hard spheres and were forced by
pressure into the closest possible packing. The positions of the atoms

are given by 1 , , > \ oy /s. i iy\ where the codrdinates refer

[fn + }4,n + %,(p + }i)c

to hexagonal axes, and w, «, />, have all possible integral values. The third

lattice in Table IV. is composed of three of the above simjple triangular

Nttd.

X-RAY CRYSTAL ANALYSIS.

6^9

lattices^ the atoms of the second and thihl being du'ectly above the centers
of the alternate trian^es df th^ first lattice, at distances of |^ and ^ re*
spiectively of the height of the i^rism. It is the regular rhombohedral
lattice. The positions of the atoms in this lattice may be most simply
specified with refei^nce to trigonal akes, but for cbnveniehce of comparison
with the first t#o latticed, they are given in teri^ of hacagdnal ax^.
Their hecsigonal Gol^itlinates arcf

Wh^r6 M, H, and p have iEill possible integral values, ahd c is t)i6 axial
ratib 1.624.
The first column in Table IV. givies the indices of the forin, the second.

Table IVi

Sliiiple TriaagnUr
Lattict.

Close-packed Lattice.

Rhombohedral Lattice.

Indices

of
P«rm.

Co-
operate

ihg
PIAnV

^2f&.^

Hum*

berof

Co-

bperat-

Pltnci.

•KfSl^'

ber of

Co-
operat-
ing
Planei.

'WeV'

Trigo-
nal In-
dices

of
Form.

0001

1.624

loTo

.866

0001

(H «- 2) .812

.812

1011

.764

100

1012

.592

110

0001

(fi ^ 3) .541

.541

111

ll2o

.500

IlO

1121

.477

1013

.458

iOio

(« « 2) .433

(n = 2) .433

ii2i

.426

20il

.418

111

0001

{n - 4) .406

(n - 2) .406

loTl

(n = 2) .382

(if = 2) .382

(n - 2) .382

100

10T4

.368

211

1123

.367

210

2023

.338

2130

.327

oool

(n = 5) .325

2l5l

.321

210

1124

.315

f 1015
12132

6
12

.304
.304

6
12

.304
.304

3
6

.304
.304

221
2II

1012

(n *= 2) .296

(« = i) .296

(« ^ 2) .296

110

1010

(n * 3) .289

.289

211

68o A, W, HULL. I

fourth and sixth the number of different families of planes belonging to
the form, and the third, fifth and seventh the spacing of these planes
in the respective lattices, found by substituting the codrdinates of the
atoms in equation (5). The unit is the side of the elementary triangular
prism. The eighth column gives, for comparison, the indices of the
planes of the rhombohedral lattice in trigonal (Miller) codrdinates.
The atoms are in this case referred to three equal axes, making equal
angles of 78.4** with each other, and their codrdinates are m, «, p, where
each of these numbers has all possible integral values.

Examples.

As examples of the application of the method of analysis described
above, the analysis of ten elementary crystalline substances is given
below. Three of these analyses are incomplete, but are of such im-
portance as to warrant their inclusion. Four others have already been
briefly described elsewhere, and are given here in more detail. The last,
diamond, which has been completely analyzed by the Braggs, is added
as a check u|x>n the method, and as an example of the immense amount
of information which can be obtained from a single photograph.

The experimental data is collected in Tables V. to XIV. The first
column in each table gives the estimated intensity of each line. The
estimate is necessarily very rough, but photometric measurements have
little value unless care is taken to make control exposures to determine
the characteristic curve of the plate under the actual conditions of
exposure and development. In the photographs here described, this
was not done. The second colunm gives the distance, x, of each line on
the photograph from the central undeviated image of the slit. The third
column gives the angular deviation 2$, of the ray that produced the line,
calculated from x and the distance between crystalline material and
photographic plate. The fourth and fifth columns give the experi-
mental and theoretical values of d/n, where d is the distance in Angstroms
between consecutive planes, and n the order of reflection. The experi-
mental values of d/n are calculated from the angular deviation 26 by
means of the equation n\ = 2d sin $. The theoretical values are ob-
tained by multiplying the values in Tables III. and IV. by the lattice
constants of the respective crystals. The sixth column gives the indices
of the forms to which the reflecting planes belong, and the last colunm
the number of families of planes belonging to the given form and having
the same spacing, so that their reflections are superimposed. The
number of these cooperating planes is a measure of the intensity of the
line to be expected if the atoms are symmetrical and equally distributed
in successive planes.

5s"^*] x-ray crystal analysis. 68 1

Iron.

The iron investigated was obtained from two sources, viz., fine filings
of pure electrolytic iron, and fine iron powder obtained by the reduction
of FeiQi in hydrogen. The filings were mounted in a thin-walled glass
tube 2 mm. in diameter, which was kept in rotation during the exposure.
The reduced oxide was pressed into a sheet 2 mm. thick, which was
moimted firmly at right angles to the beam of X-rays. Both specimens
gave the same lines.

A fine-focus Coolidge X-ray tube with tungsten target was used for
all the iron photographs. It was operated by the constant potential
equipment which has been in use for two years in the Research Labora-
tory,* at 110,000 volts and i milliampere.

Fig. 6 shows one of the photographs of the iron powder (reduced
oxide). For all lines beyond the first three, the a doublet is resolved
into two very narrow, sharp lines. The fi line of the K radiation is
visible on the plate for some of the stronger reflections, but is easily
distinguishable from the double a line. In this exposure both slits were
very narrow, about 0.2 mm. wide. The distance from X-ray tube to
first slit was 20 cm., from first to second slit 15 cm., and from crystal to
photographic plate 18.15 ^^^' ^^^ X-ray plate was used, with calcium
tungstate intensifying screen. The exposure was 20 hours.

The lines in this photograph are tabulated in Table V., together with
the calculated spacings, as described above. The observed spacings
(column 4) agree with the theoretical spacings for a centered cube of
side 2.86 A. (column 5) within the limit of accuracy of measurement of
the lines. The intensities also vary in. the manner to be expected,
except that the second order no line is too intense and the second order
ioo too weak. The bearing of this fact on the question of the arrange-
ment of electrons in the iron atom has been discussed elsewhere.*

A centered cubic lattice should have two atoms associated with each
elementary cube. By equating the mass of the n atoms in an elementary
cube to the mass of the cube, i. e., its volume X density of the metal, we
obtain

_ pd' 7.86 X 2.86* X 10-^
"* ~ Jlf ■" 554 X 1.663 X 10-" " ^*~-

As a check upon this analysis, photographs were taken of single
crystals of silicon steel, containing about 3.5 per cent, silicon, which
were mounted on the spectrometer table and rotated about definite
axes. Two of these photographs are reproduced in Fig; 7. The first,

> Phys. Rbv.. 7, 405, 1916. For a fuller description see G. E. Review, 19, 173. March. 19x6.
* Phys. Rev.. 9, 84, 191 7.

68s

A> W. BULL.

Table V.

Inm.

Ihtsnaity of

DIsUA^of

L4De from

Ctator.

Aatular
DoTiatioQ of

bp4dnc of PlAaes in

ladicoi of Form.

Cikoptratiac

ftstimati^.

cm.

bofrooa.

Bxfiefiliktii-
tal.

thtortti-
cal.

1.00

1.87

5.90

2.05

2.00

110

.46

2.67

8.40

1.43

100

.54

3.40

10.30

1.16

211

.24

3.85

11.96

1.005

1.01

110 (* «* 2)

4.32

13.40

.910

.905

310

.16

4.75

14.67

.823

.826

111

.22

5.17

15.90

.757

.765
.715

321

100 (« - 2)

.12

5.92

18.06

.665

.675

f 110 (n - 3)
1411

.03

6.27

19.06

.633

.638

210

.02

6.62

20.06

.600

.610

332

.02

7.00

21.10

.572

.584

211 (H >* 2)

.10

7.25

21.78

.555

.560

f510
\431

.02

7.95

23.16

.522 .522

521

Fig. 7a, is the photograph of a thin crystal about 5 mm. square, cut
parallel to 100. It was mounted 12 cm. from the photographic plate,
with its 100 face normal to a beam of tungsten rays, and rotated slowly,
about an axis perpendicular to 001, for a few degrees on each side of
the center. It shows, in the horizontal plane, the spectrum of the
tungsten target reflected from 010, and at 45** and 135® the same reflected
from on and oTi respectively. The two K lines in tungsten, the un-
jesolved a doublet and the fi line, show plainly in each of these spectra,
and the distance between the a doublets of the right and left spectra,
viz., 3.56 cm. for 010, and 2.50 cm. for on, give for the spacings of these
planes

doio = 1.43 A.,

don = 2.04 A.

The second photograph. Fig. 76, shows the result of rotating a thin
crystal cut parallel to in, and mounted normal to the rays, about an
axis perpendicular to no, for a few degrees on each side. It shows, in
the horizontal plane, the reflection from T12, and at 30®, 60** and 71**,
to the horizontal the reflections from lot, 211, 321, with corresponding
reflection from 231, 121 and oil at angles of 109**, 120** and 150** respec-
tively to the horizontal. The planes 7oi and oTi show both first and
second order spectra. The distances between a lines of these spectra

JS2"^] X-RAY CRYSTAL ANALYSIS. 683

agree excellently for planes belonging to the same form, and give for the
spacing of the planes in the three forms represented :

dtix = 1. 15 A.,

duo ™ 2.02 A.,

dm = 0.75 A.
The agreement of these angles and spacings with the theoretical values
for a centered cubic lattice indicates that the position of the atoms in
iron is not greatly affected by the presence of 3H pcr cent. Si.

A series of photographs of one of these crystals at liquid air tempera-
ture, room temperature, and 1000® C. respectively showed no observable
change, even in intensities. It is necessary to photograph more forms,
however, before definite conclusions can be drawn regarding the relation
of a to jS iron. Several photographs of iron f)owder at different tempera-
tures between 700** C. and 900® C. were sf)oiled, either by chemical fog
due to the heating of the photographic plate, or by the growth of the
crystals during exposure, thus giving only a few large spots on the

photograph.

Silicon.

Small crystals of metallic silicon were crushed in a mortar and sifted
through a gauge of 200 meshes to the inch. The fine powder was mounted
in a very thin-walled tube of lime glass, and kept in continuous rotation
during a four hour exposure to rays from a molybdenum target, running
at 32 k.v. constant p)otential and 8 milliamperes. A filter of zircon
powder .037 cm. thick reduced the spectrum essentially to a single line, the
unresolved a doublet, X = .712, of molybdenum, as shown in Fig. 3. The
crystal was 15 cm. from the X-ray target and 11. 3 cm. from the photo-
graphic film, which was bent in the arc of a circle, with the crystal at the
center. Both slits were quite wide, about i mm., and about 5 cm.
apart. Eastman X-ray film was used, with calcium tungstate intensi-
fying screen.

The photograph obtained is reproduced in Fig. 8, and the measure-
ments are given in Table VI. The spacings tabulated as "theoretical"
are those of a lattice of the diamond type, i. e., two intermeshed face-
centered lattices, each of side 5.43 A., one lattice being displaced, with
reference to the other, along the cube diagonal a distance one fourth
the length of the diagonal. The agreement is perfect. The estin^tes
of intensity are not accurate enough to warrant discussion.

The number of atoms associated with each unit cube is

P(P 2.34 X 543' ^^
^^ M ^ 28.1 X 1.663 ~ ^'^'

which is the correct number for this type of lattice.

684

A. W. HULL.

Table VI.

Silicon,

Intensity of
Une.

Distnnc* of

Lino from

Center.

Anfulnr

Dorintion of

Lino li.

Spacing of Pianos in
Angstroms.

N amber of

Indicss of Form.

Cooperating
Pinnes.

B«tinuit«d.

Cm.

Dogrooo.

Bzporimsn-
tnl.

Thooroti-
cal.

1.00

2.58

13.56

3.13

3.14

Ill

.80

4.21

23.0

1.93

110

.75

4.97

26.16

1.64

311

•

1.57

111 (» -2)

.25

6.00

31.58

1.36

1.356

100

.45

6.54

34.42

1.25

331

.50

7.39

38.92

1.11

211

.40

7.84

40.78

1.05

1.04

f511

\lll (11-3)

.20

8.59

45.20

.96

110 (« -2)

.30

8.98

47.22

.92

531

.25

9.66

50.44

.86

310

.10

10.01

52.44

.83

533

.82

311 (n - 2)

.05

10.62

55.70

.79

111 (» « 4)

.10

11.00

59.76

.76

(711
1551

.20

11.58

60.36

.73

321

.15

11.90

62.56

.71

f731
1553

.68

100 (» « 2)

.66

733

.05

13.33

70.0

.64

411

.05

13.60

71.44

.63

(751

I 111 (ii-5)

Aluminium.

Fine filings of pure sheet aluminium were mounted in exactly the
same manner as silicon, and exposed for 3 hours to molybdenum rays,
produced at 40,000 volts, 9 milliamperes, and filtered by .037 cm. of
zircon powder. The photograph obtained is shown in Fig. 9, and the
measurements are given in Table VII.

The "theoretical" spacings in Table VII. are those of a face-centered
cubic lattice. Their agreement with the "experimental *' values obtained
from the lines on the photograph is satisfactory.

The number of atoms per unit elementary cube is

_ pd» 2.70 X (4>05)' .^
^ ■" Jlf " 2.69 X 1.663 " ^

This is the correct number for a face-centered lattice.

Na6. J

X-RAY CRYSTAL ANALYSIS.

Table VII.

Aluminium,

685

lBt«nsity
of Lin*.

Distance of

Line from

Center.

Angular

Deviation of

Line fl $,

Spacing of Planes in
Angstroms.

Indices of Form.

Number of

Coi>perating

Planes.

Bstimatcd.

Cm.

Degrees.

Bxperi-
mental.

Theoretical.

1.00

3.45

17.80

2.33

Ill

.60

3.99

20.60

2.025

100

.50

5.67

29.26

1.43

.60

6.68

34.5

1.21

1.22

311

.20

6.95

35.8

1.17

111 (« « 2)

.05

8.09

41.8

1.01

100 (n = 2)

.25

8.86

45.6

.93

331

.25

9.11

47.0

.90

210

.10

10.05

51.8

.82

.83

211

.15

10.66

55.0

.78

(511

I 111 (« - 3)

.02

11.75

60.6

.71

.72

110 (n - 2)

.04

12.30

63.4

.68

531

The unit of structure of the aluminium crystal is, therefore, a face-
centered cube, of side 4.05 A., with one atom of aluminium at each
comer and one at the center of each face.

Magnesium.

The magnesium used in these experiments was the commercial elec-
trolytic product made in the research 'laboratory. Several photographs
were taken, some with fine filings from cast rods of this metal, and some
with filings from large crystals formed by vacuum distillation. Both
kinds of powder gave the same results.

The powder was mounted in a 2 mm. tube of thin glass, and exposed
under exactly the same conditions as silicon and aluminium. Fig. 10
shows a photograph obtained from a 6-hour exposure at 32,000 volts
and 9 milliamperes, and Table VIII. gives the numerical data.

The "theoretical spacings" in Table VIII. are those of a hexagonal
lattice composed of two sets of triangular prisms, each of side 3.22 A.
and axial ratio 1.624, with construction points 000 and 3^, %, J^ respec-
tively. This is the lattice whose spacings are given in column 5 of Table
IV., under ** Close-Packed Lattice." It is slightly distorted, however,
from true hexagonal close packing, which requires an axial ratio of 1.633.
This variation from theoretical close packing is to be attributed to a
slight asymmetry in the structure of the magnesium atoms.

The agreement between calculated and experimental spacings is satis-
factory, except that several lines which were to be expected do not

^86

A, W, HULL.

show in the photograph. In particular, the reflection from the basal
plane, oooi, is absent in all the photographs.

It seemed desirable, therefore, to supplement the evidence furnished
by the powder photographs by photographs of single crystals, mounted

Table VIII.

Magnesium.

Intensity of
Une.

Distance of

Line from

Center.

Anfleof
Reflection.

Spndnff of Planes in
Angstroms.

Nnmber of

Indicee Qf Form.

CooperaUns

BsUmated.

Qm.

Dsfrses.

Experimen-
tal.

Theoreti-
cal.

.40

2.92

14.80

2.75

2.59

loTo

0001

1.00

3.30

16.66

2.44

lOU

.30

4.23

21.48

1.91

1.90

1012

.40

5.03

25.50

1.61

1.60

1120

.35

5.50

27.8

1.48

1.48
1.38

1013
10T0(2)

6
3

.35

5.95

30.0

1.36

1132

.12

6.05

30.6

J. 34

1.34
1.30

2021
0001(2)

.06

6.65

33.6

1.23

10Tl(2)

.02

6.9

34.8

1.18

1014

.10

7.52

38.0

1.09

1.08

2023

.18

7 96

40.2

1.04

1.05

2130

.02

8.1

41.0

1.02

1.03
1.01

2131
1174

12
6

.1?

8.41

42.6

.98

.97
.94

r2132
11015
1012(2)

18
6

.01

8.83

44.8

.93

.92

1010(3)

.10

9.15

46.2

.90

.89

2133

.06

9.48

48.0

.87

f3032
10001(3)

9.90

50.2

.83

.82
.80

f 1016
12025
f 10Tl(3)
12134
1120(2)

.06

10.95

55.4

.77

3140

with definite orientations. Several such photographs were taken, the
measurements of three of which are given in Table IX. The crystals
were formed by vacuum distillation, and were about 2 mm. in diameter.
The first was mounted with its basal plane (oooi) parallel to the rays,
and rotated slowly about an axis normal to 1210, for about 30^ on ^^
side of the center.^

1 The reflection from zozo ihould not have appeared on tliis plate. It w^ very f^t, bat
clearly visible on both sides, and was probably due to a twin upon zoii, a small portion

VOL.X.1

No. 6. J

X-RAY CRYSTAL ANALYSIS.

687

The second crystal was mounted so as to rotate about the same axis
as the first, but with loTo parallel to the rays at the start. The third
was mounted with 1120 parallel to the rays, and rotated about an axis
normal to iioo. The patterns obtained in these photographs differed
from those of the single iron crystals in containing reflections from
many more planes, corresponding to the greater complexity of the
hexagonal system. Molybdenum rays were used, filtered through .037
cm. of zircon, so that the spectrum consisted of a single line.

The lines reflected in the horizontal plane, which appeared on the

three photographs, are given in Table IX. The 0001 reflection was

very strong on the first photograph, and on three additional photographs

which were taken to make certain its identity. Its absence in all the

powder photographs must be due, therefore, to the much greater relative

intensity of the reflections from other forms, containing many more

planes. These forms reflect not only the lines but the unabsorbed part

of the general spectrum, causing a fog over the plate that obscures weak

lines.

Table IX.

Ciystmlz.

Crystal s.

Crystal 3.

Position
of Line.

Spacing
of Plane.

Indices
of Plane.

Position
of Line.

Spacing
of Plane.

Indices
of Plane.

Position
of Line.

Spacing
ot Plane.

Indices
of Plane.

3.10
2.90

3.30
4.20
5.50
6.27

2.59
2.75
2.44
1.90
1.48
1.30

0001

loTo

lOll
IOT2
1013
0001(2)

2.90
3.10
3.30
6.05
8.92
5.9

2.75
2.59
2.44
1.34
0.92
1.38

loTo

0001

lOlX
2021
10T0(3)
10T0(2)

5.02
5.50
6.0

1.60
1.48
1.36

li5o

1021
1122

The lines tabulated in Table IX., and all the others which appeared
on these photographs, are the ones which should appear, with the excep-
tions mentioned in the above note.

The evidence seems sufficient that the assumed structure is correct,

viz., that the atoms of ms^^esium are arranged on two interpenetrating

lattices of triangular prisms, each of side 3.22 A. and height 5.23 A.,

with one atom at each corner, the atoms of one set being in the center

of the prisms of the other.

Sodium.

The first photographs of sodium were of rods, about i mm. in diameter
and I cm. long, cut from an old sample that had been in the laboratory

of which was included in cutting the crystal from the mass of other crystals upon which
it grew. A second specimen mounted and photographed in the same manner did not show
this line. Similar twinning must account for the 0001 reflection shown by crystal a, and
X013 by crystal 3.

688

A. W. HULL.

several years. These rods were placed in sealed glass tubes, and exposed
to molybdenum rzys. They gave intense reflections, of a pattern which
indicated that the lump from which the samples were cut was a single
large crystal.

Several unsuccessful attempts were then made to obtain finely divided
crystals of sodium. Distillation in vacuum into the thin-walled tube
which was to be photographed was found impossible. Several different
glasses and pure silica tubing were tried. The sodium always ate through
the tube wall l)efore it could be coaxed into the narrow tube. Melting
the distilled sodium so that it flowed into the tube resulted in an amorph-
ous condition, which gave no lines at all. It is probable that distillation,
had it succeeded, would have given the same result, for potassium distilled
in this way was found to be completely amorphous. Shaking in hot
xylol gave a beautiful collection of tiny spheres, but these too were
amorphous, and annealing for i6 hours at 90^ C. failed to produce any
appreciable crystallization. Crystallization from ammonia solution gave
a black mass, from which it was diflficult to separate the pure sodium.
Fairly good photographs were obtained with fine shreds, scraped from
the lump, with a knife, under dry xylol, and packed in a small glass tube.

A satisfactory sample was finally prepared by squirting the cold
metal through a .01 cm. die, and packing the fine thread, with random
folding, into a i mm. glass tube, which was immediately sealed. The
sample from which this was taken was about two months old, and was
apparently only slightly crystallized, so that only a few lines were visible,
on the dense continuous background due to the amorphous part. Two
photographs, taken under the same conditions as the preceding, with
exposures of 4 and 14 hours respectively at 30 k.v. 27 milliamperes, gave
identical lines, which are tabulated in Table X.

Table X.

Sodium.

Intensity of
Line.

Distance of

Line from

Center.

Angular

Deviation of

Line a«.

Spacing of Planes in
Angstroms.

Indices of
Form.

Number of

Cooperatiag

Flanea.

Bstimated.

Cm.

Degrees.

Experi-
mental.

Theoretical

(Centered

Cube).

1.00
.10

.40
.10
.08
.02
.05

266

3.78
4.66
5.38
6.03
6.57
7.18

13.38
19.06
23.36
27.04
30.26
33.40
36.0

3.05
2.15
1.76
1.52
1.36
1.24
1.15

3.04
2.15
1.76
1.52
1.36
1.24
1.15

110

100

211

110(2)

310

111

3
12

6
12

4
24

VOL.X.

Na6

^'] X-RAY CRYSTAL ANALYSIS, 689

These seven lines, which are the only ones that appeared on any of
the sodium photographs, agree perfectly with the theoretical spacings of
a centered cubic lattice, of side 4.30 A., and cannot be made to fit any
other simple type of lattice.

The number of atoms per elementary cube is

6.970 X 4^

22.8 X 1.663 ^

which is as close to the required number, two, as the data would warrant.

The evidence is sufficient, I think, in spite of the limited number of
lines, to show that the atoms of sodium, when in its crystalline form,
are arranged on a lattice whose unit of structure is a centered cube, of
side 4.30 A., with one atom at each corner and one in the center of the
cube. The tendency to form this regular arrangement is, however,
very slight, corresponding to a small difference between the potential
energies of the crystalline and amorphous states. This fact is important
for the determination of the structure of the sodium atom.

The structures of the elements thus far described have been deter-
mined with considerable certainty. The three following have been
only partially determined, but are included as examples of the possi-
bilities, as well as the difficulties, of the analysis.

Lithium.

The structure of lithium is of special interest because, on account of
the small number of electrons associated with each atom, it may be
expected to yield valuable information regarding the arrangement of
these electrons around the nucleus. The analysis is difficult, however,
on account of the complete lack of crystallographic data, the slowness
of crystallization, and the difficulty of obtaining pure metal.

It was first attempted to distill lithium in vacuum, for the double
purpose of purification and of obtaining small crystals. Various methods
of heating the metal were tried, such as a tungsten spiral with the lithium
ribbon lying in its axis, a molybdenum cup heated externally by electron
bombardment, etc., but without success. The metal reacts violently
with glass and silica at temperatures far below those at which its vapor
pressure is appreciable.

Two samples were used. The first was prepared by electrolysis of
pure lithium chloride, in a graphite crucible, and probably contains
little impurity except carbon. A small lump was rolled, between steel
surfaces, into a cylinder 2 mm. in diameter, and sealed in a glass tube.
It was exposed, in the same manner as the previous crystals, for 7 hours

690

A. W. HULL.

to molybdenum rays at 40 k.v. and 6 milliamperes, and gave the lines
tabtdated in Table XI.

Table XL

Lithium.

IntMksitj of
Line.

Distance of

Line from

Center.

Ancolnr

Dcriirtion of

Lineal.

Bpncincof Planes la
Ani^stroflM.

Indices of

Bstinuited.

Cm.

Dcfreee.

Bxperi. Theofeljcnl

mentaL <?}!"Pl«
Cube).

Cobpcratlac

3.50

100

.70

3.20

16.50

2.50 2.48

1.00

3.96

20.44

2.02 2.02

111

.05

4.61

23.4

1.75

1.75
1.56

100(2)
210

3
12

.40

5.62

29.0

1.43

211

.02

6.5

33.6

1.24

110(2)

.60

6.95

35.9

1.17

1.10
1.05

f221
1 100(3)

310

311

15
12
12

.10

8.07

41.6

1.01

1.01
.97

.87

.85

111(2)

320

100(4)
f410
1322

.05

9.99

51.6

.83

.83
.80

f411
1110(3)
331

18
12

.20

10.82

56.0

.77

.78

210(2)

The spacings calculated are those of a simple cubic lattice, of side
3.50 A., and the density of lithium requires that 2 atoms be associated
with each point of the lattice, viz. :

n =

Af ^6.89 X 1.663

= 2.00.

A centered cubic lattice in which half of^the atoms, those belonging to
one of the two component simple cubic lattices, are oriented oppositely
to the other half, could probably be made to fit the observations by
assuming a suitable arrangement of the electrons in the atoms. It is
more probable, however, in view of the next photograph, that the
strong lines at 3.96 cm. and 6.95 cm. are due either to an impurity or
to the admixture of a second form of lithium. All the other lines in
Table XI. are consistent with a centered cubic lattice, of side 3.50 A.,
with one atom of lithium at each cube comer and one in the center of
each cube.

Na6. J

X-RAY CRYSTAL ANALYSIS.

691

The second sample was taken from a very old stock of supposedly
very pure lithium, origin not known. A fine thread was squirted through
a die and packed into a glass tube, in the same manner as sodium.
A five-hour exposure to molybdenum rays, at 30 k.v. 27 milliamperes,
gave only 3 lines, viz., a strong line at 3.22 cm., and two weaker lines
at 4.60 cm., and 5.46 cm. These are exactly the positions of the first
three lines of the centered cubic lattice described above, and it is espe-
cially noteworthy that the line at 4.60 is relatively much stronger than
on the preceding photograph, and the strong lines at j,g6 cm. and d.95
cm. are entirely lacking. One is tempted to consider this last photograph
as that of pure lithium, since its interpretation is simpler than the pre-
ceding, and it gives to lithium the same structure as sodium. The num-
ber of lines is too small, however, to justify this conclusion, and further
experiments with purer metal are needed.

Nickel.

Specially purified nickel wire was melted in vacuum and cast in a
lump. Filings from this lump were placed in a small cell 2.5 mm. thick,
and exposed 4 hours to tungsten rays, produced at 110,000 volts, i
milliampere, filtered through .015 cm. of tantalum. The photographic
plate was placed 15.7 cm. from the crystal, at right angles to the beam of
X-rays. The lines obtained are tabulated in Table Xll.

Table XII.

Nickel.

Intensity of
Line.

DistanceTof

Line from

Center.

Ani^ular

Deviation of

Line 9$.

Spacing of Planes in
Angstroms.

Estimated.

Cm.

Degrees.

Experimen-
tal.

Theoretical

(Centered

Cube).

Very strong

Faint

1.70
2.42
2.97
3.42
3.80

4.58

6.17

8.75

10.70

12.25

13.60

16.28

1.95

1.38

1.13

.98

.89

.74

1.95
1.38
1.13

.98

.87

.79

.74

Strong

Medium

Faint

Strong

Number of

Indices of
Form.

Coooerating

110

100

211

110(2)

310

111

321

The spacings agree perfectly with those of a centered cubic lattice,
of side 2.76 A., with one atom of nickel at each cube corner and one in the
center of the cube. Taking the density of pure nickel as 9.00, which
is probably too low, the number of atoms associated with each elementary
cube is

692 A. W. HULL,

p^ 9.00 X 276t
^^ M "5^.2X1.663" '-^5.

which is as dose to the required value, 2, as the data will warrant.

Three other photographs, one of a thick electrolytic deposit on very
thin nickel foil, the other two of a 2 mm. nickel rod of unknown origin,
gave quite different lines. The electrolytic deposit was exposed but a
short time, and gave 4 lines, at 1.64, 1.89, 2.70 and 3.14 cm. corresponding
to spacings of 2.01, 1.76, 1.25, 1.07 respectively, vohidi art exactly the
spacings of the first four Unes of a fau<entered cube, of side 3.52 A. The
number of atoms associated with the elementary cube is

9X3^ _
58.2 X 1.663 " *-^'

which is correct for a face-centered cubic lattice cont^uning one atom
of nickel at each cube comer and one in the center of each face.

The other two photographs, of the nickel rod of unknown origin,
contained the lines of both the preceding ones, but only these lines. It
was presumably a mixture of the two crystalline forms of nickel, repre-
sented by the two preceding specimens respectively.

The evidence is very strong, therefore, that nickel crystallizes in two
different forms, ope a centered cubic lattice, like iron, and the other a
face-centered cubic lattice, like copper. The relation of the magnetic
and mechanical properties to these crystalline changes has not been
studied, and the above analysis is to be regarded as only preliminary.

Graphite.

Several photographs of both natural and artificial graphite have
been taken. The natural graphite was in large flakes, obtained from the
Dixon Crucible Company. The artificial graphite was a very fine
powder, furnished by the Acheson Company. Both had been heated to
3500** C. in a special graphite furnace to remove impurities and ash.

The natural graphite, either in large flakes or where pressed into a
glass tube, gave very unsymmetrical photographs, showing the pre-
dominance of certain orientations of the crystals. By forcing it through
a copper gauze of 100 meshes to the inch, a powder was obtained which,
when packed in a glass tube and kept in rotation, gave very r^^ular and
symmetrical photographs. The Unes in these photographs were identical
with those in the photographs of artificial graphite, showing that the two
are identical in crystalline structure. One of these photographs, obtained
from a 16-hour exposure to Mo rays at 34,000 volts and 16 milliamperes,
is reproduced in Fig. 11, and the lines, together with the calculated
spacings, are tabulated in Table XIII.

No. 6. J

X-RAY CRYSTAL ANALYSIS.

693

Table XIII.

GraphiU.

Intensity of
Line.

Distance of
Center.

Anfular

Deviation of

Line.

spacing of Planes in
Angstroms.

Indices of
Form.

Nomber of

Cooperating

Planes.

Betimated.

Cm.

Degrees.

Bitperimen-
tal.

Theoretical.

100

2.40

12.16

3.37

0001

3.84

19.46

2.11

2.12

lOlS

3.99

20.20

2.03

2.02

1011

4.47

22.60

1.81

1.80

1012

4.81

24.34

1.690

1.685

0001(2)

5.21

26.38

1.560

1.544

OOT3

•

1.318

1014

6.65

33.70

1.227

1120

7.09

35.90

1.155

1.152
1.138
1.124
1.062

1152

loiS

0001(3)
1010(2)

6
6

1
3

7.82

39.60

1.050

1.048
1.008

20?1
10Tl(2)

6
6

8.31

42.10

.990

r.994
1.990
.960
.897
.877
.842
.833

IOT6

1154

2023

10l2(2)

IOT7

0001(4)

202S

6
6
6
6

1
6

10.06

51.0

.827

.829

1126

10.41

52.8

.800

r.802
1.797
.783
.780
.773
.756
.725
.715

2130

2131

IOT8

2132

1013(2)

2153

2134

2027

6
12

6
12
12

11.85

60.0

.712

r.708
1.708

1010(3)
1019

3
6

12.11

31.4

.697

/.696

1.693

.690

1128
3032
2135

12.61

64.0

.672

/.674

1.674

.660

10ll(3)
0001(5)
10l4(2)

1
6

12.94

65.6

.656

r.654

1.654

.644

303i

2136

10110

13.79

69.8

.621

r.616
1.616

2137
ll50(2)

12
3

14.11

71.50

.609

r.612
1.612

2029
2241

6
6

694 ^' ^' B^^^

XIII

I .603 1121 '2y 6

3W 1012 3, 6

14^7 '73-8 J92 1-*^ ^ ^

14J^7 73-« .592 1^^ j^jj^ ^

The crystanos:rairfuc data regarding graphite is very meager and un-
certain, and in attempting to guess its crystalline structure one has an
embarrassing freedom of choice, both of crystal systems and of axial
ratios and angles. The only guidii^ principles, apart from the lines in
the photograph are, first, that the true structure is probably very simple
and symmetrical, suice all its atoms are alike, and second, that the nearest
approach of adjacent atoms cannot be very different from that in
diamond.

The structure whose spacings are tabulated in Table XIII. fits the
experimental data best of all that have been tried, and seems capable,
when account is taken of the internal structure of the atoms, of explainii^
all the observed intensities of the lines. It is a hexagonal structure,
composed of four simple lattices of triangular prisms, each of side 247
A. and height 6.80 A., the atoms of the third lattice being directly above
those of the first at a distance of one half the height of the prism, those
of the 2d and 4th lattices being above the centers of alternate triangles
of the first, at distances 1/14 and 8/14 respectively of the height of the
prism. The codrdinates of the atoms are:

m'+li, n+H, (p+ifi4)c,

fn, n, (p + yQc,

m + %, n + H, (p + 8/i4)c,

where m, n, and p have all possible values and c, the axial ratio, is 2.75.
The 0001 planes are thus arranged in pairs, similar to the iii planes in
diamond. The distance between nearest consecutive planes, and be-
tween atoms in each plane, 48 A. and 247 A. respectively, are slightly
less than their values .51 and 2.52 for diamond, and the nearest approach
of atoms is 1.50 A. as compared to 1.54 for diamond. This closer ap-
proach of the atoms in graphite would indicate chemical stability. The
distance between consecutive pairs of planes, however, is much greater,
viz., 3.40 A. in graphite, than its value 2.06 A. in diamond, which accounts
for the extreme ease of basal cleavage and gliding in graphite. |^

JJJJ-^] X-RAY CRYSTAL ANALYSIS. 695

The agreement between experimental and calculated spacings in Table
XIII. IS well within the limit of the experimental error, which is about
I per cent. Every experimental spacing is accounted for, the first 12
with certainty, the last 7 with some ambiguity on account of the large
number of theoretical spacings. The absence of reflection from planes
such as 10T4, 10T5, the second orders of loTo and loTi, and the third and
fourth orders of 0001, is dependent not only on the positions, but on the
internal structure of the atoms, and cannot be interpreted except in
conjunction with a study of this internal structure, which will be under-
taken as soon as accurate photographic measurements can be obtained.

The structure given above has the lowest synmietry of any elementary
substance yet studied. It may be that the essential elements of the
hexagonal lattice can be more simply represented by a monoclinic or
triclinic, or possibly an orthorhombic lattice, though efforts in this
direction have so far been unsuccessful.

Diamond.

The crystal structure of diamond has been completely determined by
the Braggs,^ and confirmed by nunlerous observers. Comparison of the
results of these investigators with those obtained from a powder photo-
graph will therefore serve as an excellent check upon the latter. In
addition, the powder photograph of diamond has a merit of its own, for
it furnishes evidence not hitherto available regarding the internal struc-
ture of the most interesting of all atoms. The photographs taken thus
far are not suitable for photometering, but arrangements are complete
for taking such photographs, and for measuring the intensity of the lines.

Several photographs of diamond have been taken, under varying
conditions, with identical results, as regards position and relative inten-
sity of lines. Fig. 12 shows the result of a fifteen -hour exposure to Mo
rays at 30,000 volts, 35 milliamperes, with zircon filter of 0.37 mm. A
very thin wall glass tube of special lithium boro-silicate glass, 2 mm. in
diameter, was filled with diamond powder, obtained by crushing some
old dies in a steel mortar. This powder was mounted on the spectrom-
eter table, concentric with a wooden disc 10.27 cm. in diameter, upon
which Elastman X-ray film was fastened in a complete circle, except for a
5 mm. hole where the rays entered. The collimator slits were about 1.5
mm. wide, and the distance from X-ray target to powder was approxi-
mately 35 cm. Only one half of this film, corresponding to angles of
diffraction from o** to 180**, is shown in Fig. 12. Twenty-five of the
possible 27 lines are visible in the photograph, the last two being obscured
by the dense fog.

> X-Rays and Crystal Structure, p. X02 ff., Proc. Roy. Soc. A., 89, 277.

696

A. W. HULL.

Table XIV.

Dimmtmd.

Intcailty of
Line

Dlstmnccof
Liacfrom
Center X

Angular

Dcvtetiooof

Lineai.

apTJngof PImmsIb
Aafstronw.

ladicMof
PenB.

NiuBbcr of

FlADOS.

«» -* 1

Batimstcd.

Cm.

DeffTMS.

«P2- , TlMorcticaL

1.00

1.80

20.06

2.05

2.06

Ill

.50

2.96

33.0

1.26

110

.40

3.49

39.92

1.072

1.075

311

.10

4.26

47.4

.885

.890

100

.25

4.66

52.0

.813

.817

331

.40

5.31

59.2

.721

.728

211

.20

5.66

63.0

.680

.683

/ 111(3)
1511

.10

6.22

69.4

.625

.630

110(2)

6.54

73.0

.597

.602

531

.15

7.10

79.2

.558

.563

310

.06

7.43

82.8

.538

.543

533

.03

7.98

89.0

.507

^13

111(4)

.08

8.24

91.8

.496

.498

f711
1551

.20

8.76

97.6

.473-

.476

321

.15

9.06

101.0

.462

.463

r731
1553

.005

9.70

107.6

.442

.445

100(4)

.003

10.00

113.2

.432

.435

733

.12

10.52

116.8

.417

.420

Ull
1 110(3)

.08

10.84

120.8

.409

.411

r751
1111(5)

.05

11.50

127.6

.397

210

r.08
1.02

11.84
11.93

132.0 \
132.8 /

.389

.391

r753
1911

r.05
1. 01

12.54
12.70

139.8 \
141.4/

.378

.379

332

f.05
1.01

13.00
13.23

145.0 \
147.2 /

.372

.373

931

f.07
1.02

14.00
14.27

156.0 \
159.0 /

.363

211(2)

933

r.2o

\.06

14.83
15.35

165.4 \
171.2/

.358

755
771
311(3)

The lines and the corresponding spacings are tabulated in Table XIV.,
and compared with those required for the lattice which has been assigned
to diamond by the Braggs. The agreement is absolute. It will be
noted that for the larger deviations the doublet of the molybdenum
radiation is clearly resolved.

No'ef'] ^^^^ FREQUENCY X-RAYS. 697

THE CRITICAL ABSORPTION OF SOME OF THE CHEMICAL
ELEMENTS FOR HIGH FREQUENCY X-RAYS.

By F. C. Blakb and William Duanb.

IT has been known for a long time that marked increases in the absorp-
tion of X-rays by a chemical element take place at frequencies close
to the frequencies of the characteristic X-rays of that element. The
coefficient of absorption of the element is much greater on the high-
frequency side of the characteristic X-radiation than on the low frequency
side. In the K series of the characteristic rays of an element the a
lines are much stronger than the /3 and 7 lines, but the frequencies of the
fi and 7 lines lie above those of the a lines. It would be natural to
suppose that a marked change in the absorption would occur near the
frequencies of the a lines, for most of the energy of the characteristic
rays is radiated in these lines. Such, however, does not appear to be
the case. The curves representing the relation between the coefficient
of absorption of a few of the elements and the frequency of the X-rays
presented by one of us^ to the American Physical Society in October,
1 91 4, showed that marked increases in the absorption occurred at
frequencies considerably above those of the a lines and near those of
the /3 lines. Subsequently Bragg* made some more accurate measure-
ments of the absorption of X-rays by different elements, and came to
the conclusion that the critical absorption frequency lay at or above that
of the 7 line in the K series. The 7 line has a frequency about i per
cent, higher than that of the fi line.

Marked increases in the absorption of the X-rays by a chemical
element also occur in the neighborhood of the L characteristic lines of
the element. Here, however (as de Broglie has shown), there are three
characteristic frequencies at which sharp changes in the absorption occur.
These appear to correspond to the three critical emission frequencies
recently observed by Dr. Webster and Dr. Clark.*

We recently made an accurate measurement of the value of h by means
of X-radiation.* One of the chief sources of error (amounting in some

> William Duane, The Relations between the Wave-Length and Absorption of X-Rays.

' Phil. Mag., March, 1915. p. 407.

■ Physical Rbvibw, June, 1917, p. 571.

* Blake and Duane. Phys. Rev., Dec 191 7. p. 624.

698 F. C. BLAKE AND WILLIAM DUANE, [^SS

cases to 2 per cent.) which we found in measuring the X-ray wave-lengths
arose from the penetration of the X-rays into the crystal. The correction
for this penetration must be made, if the method of using the X-ray-
spectrometer involves the measurement of the angle made by the reflected
beam with the zero line or with its position on the other side of the zero.
In the methods in which the positions of the reflected beam are deter-
mined by the marks it makes on a photographic plate this correction
must be made. It must be applied, also, in some, but not necessarily all
of the ways of using the spectrometer in which the- ionization currents
due to the reflected rays are measured, as Blake and Duane pointed
out (1. c).

If the square roots of the frequencies of the characteristic lines in the
K series of different elements are plotted against the atomic numbers of
the elements, the points lie on curves which in certain regions approximate
to straight lines. They become markedly curved, however, in the neigh-
borhood of the K radiation of bromine. The plots published by Moseley
in his classical paper on this subject indicate this curvature very clearly.
It appeared from our work on the value of h that the absorption of
X-rays by the crystal itself, if not corrected for, would produce a curva-
ture of these plots in the observed direction and of about the observed
order of magnitude. Partly on this account, and partly on account of
the great importance of measuring the highest frequencies that are
known to be characteristic of the elements as accurately as possible, we
undertook the research recorded in this paper.

We began by measuring the K characteristic absorption frequencies
of the elements from bromine to cerium. The characteristic rays of
cerium have frequencies that are nearly as high as the maximum fre-
quencies of the X-rays that we can produce by means of the storage
battery of 20,000 cells, which we have used to generate the X-rays.
In measuring the frequencies we have employed the spectrometer which
we used in measuring the value of A, when we obtained the value of
A = 6.555 X 10-^.

Since in this work we have to measure X-rays of widely different wave-
lengths, and, therefore, of widely different coefficients of absorption,
and, since the correction for the penetration into the crystal depends
upon the wave-length, we adopted the method of using the spectrometer
which gives readings that are independent of this penetration. In this
method the X-rays pass through two narrow slits in lead disks before
they reach the reflecting crystal of the spectrometer. These slits deter-
mine the breadth of the beam of the X-rays, and, therefore, the variation
in the wave-lengths of the rays in the beam reflected from the crystal.

JJ®J~^'] HIGH FREQUENCY X-RAYS. 699

A third slit lies in front of the ionization chamber, and must be broad
enough to allow the entire reflected beam of X-rays to enter the chamber.
Evidently in this case the ionization current does not depend upon the
position of the ionization chamber, provided, only, that the entire X-ray
beam passes through the slit. The angle between the two positions of
the crystal planes for reflection on both sides of the zero line gives us
twice the glancing angle of incidence, 9, which is used in calculating the
wave-lengths by means of the formula

X = 2a sin ^ = 6.056 X lO"* sin 6 cm.

If we make measurements of the ionization current for different
settings of the crystal, and if we plot these currents as a function of
the readings of the verniers, we get a curve that rises from zero at a
certain point, reaches a maximum and then descends again. Such
a curve representing the general X-radiation was published in the original
paper by Duane and Hunt,^ in which they showed that the point at
which the curve begins to rise fulfills the quantum relation

Ve = hv,

V being the constant potential applied to the tube, e the elementary
charge, h Planck's action constant and v the frequency.

If we place a thin sheet of some chemical element in the path of the
X-ray beam, the ionization current corresponding to every position of
the crystal will be reduced. At a certain angle, however, corresponding
to the characteristic absorption of the element a marked change in the
ionization appears; for at frequencies above this characteristic frequency
the thin sheet of the element absorbs much more of the radiation than
at frequencies below it. Further, if the gas in the ionization chamber
happens to have a characteristic X-ray frequency in the region covered
by the curve, there will also be a marked change in the ionization current
at the angle corresponding to this frequency: for, as is well known,
ionization produced by X-rays having a frequency just above the char-
acteristic lines is much greater than that due to X-rays having a frequency
just below it.

The curve of Fig. i represents the ionization current in methyl iodide

as a function of the reading of one of the verniers attached to the crystal

table, and therefore, approximately of the wave-length. In this case

a thin sheet of antimony was interposed in the path of the beam. The

zero reading lies to the left of the portion of the curve shown. Near the

angle 286® 50' the curve begins to rise and this is the point for which

the quantum relation holds

Ve = hv.

> Duane and Hunt. Phys. Rev., Aug., 191 5. p. x66.

700

F, C, BLAKE AND WILUAM DUANE.

As the reading of the vernier decreases corresponding to an increase in
the angle measured from the zero, the wave-length of the reflected beam
of X-rays increases, and the curve rises, indicating that X-rays of these
wave-lengths are produced. As we pass across the frequency of the
characteristic rays of iodine a sharp drop occurs in the ionization current.
The X-rays of longer wave-lengths than this characteristic wave-length
do not excite as much ionization as those of shorter wave-lengths. Pro-
ceeding further we come to the point representing the critical absorption
of antimony. Here there is a marked increase in the value of the current.

•^ ^»' ^•' #•' «•• z** •»

~9p TP~yP ? m»' ^tf* »mr tt*
Tt^mdtng 0/ Verfitmr Af».J,

Fig. 1.

because X-rays of longer wave-lengths than this characteristic wave-
length are not absorbed as much as those of shorter wave-lengths.

Evidently these marked changes in the ionization current furnish a
means of accurately estimating the characteristic absorption frequencies
of the elements for X-rays.

In practise we did not measure the angle from the zero of the instru-
ment but we took curves on both sides of it, and measured the crystal
table angles from one side to the other, thus eliminating the determination
of the zero and increasing the accuracy of the measurements.

Fig. 2 represents a number of such measurements. The readings
were taken in the immediate neighborhood of the characteristic absorp-
tion. The two slits between the X-ray tube and the spectrometer
crystal were so narrow, and consequently the range of the wave-lengths
in the beam was so small that the entire change in the ionization current
representing the change in absorption took place within a variation of the
angle d of less than 3' of arc.

VOL.X.1

Na6. J

HIGH FREQUENCY X-RAYS.

701

We have assumed that the center of the rapidly rising portion of the
curve corresponds to the characteristic absorption for the X-rays at the
center of the X-ray beam. By drawing curves on a sufficiently large
scale we think that we have been able to determine the angle correspond-

Zirc0tiimm

mUdimm

0mC»0'SS'

#-- #*-##'-fr'

— **—

Y*y 7' so

j^A.

Fig. 2.

ing to the center"'of this line to within about one sixth of i'. This repre-
sents an error of one part in 1,100 about for barium, of one part in 2,300
about for zirconium, etc.

The table contains the results of our measurements for all the known
elements between bromine and cerium, both inclusive, with the exception
of the two gases xenon and krypton.

The values for bromine and iodine were obtained not by using bromides
and iodides as absorbing materials, but by putting ethyl bromide and
methyl iodide in the ionization chamber and measuring the ionization
current without any absorbing material. The fact that the values ob-
tained fall accurately on the curve proves that this is a legitimate method
of procedure within the limits of error of our measurements.

The curve of Fig. 3 represents the square root of the characteristic
absorption frequency as a function of the atomic number plotted from
our measurements. The values obtained by de Broglie ^ and by Wagner*
are also plotted in the figure. We think that the difference between
de Broglie's values and ours is due to the fact that no corrections for the
penetration of the rays into the crystal are necessary in our method of
measurement. In de Broglle's method, on the other hand, in which

> Comp. Rendu, July» 19x6, p. 87.
* Ann. d. Phys. 46, 1915, p. 868.

.Jj

1- - . .—
t

— .3

r^r J

- "fc

•--'.

• ■»-

c — •

9 <*

•*■

v^-

ffA

-W

r'^ 1

VOL.X.

No. 6

^] HIGH FREQUENCY X-RAYS. 703

the wave-length is calculated from the angle made by the reflected beam
measured to a line on a photographic plate, such a correction must be
made (at least for X-ray wave-lengths lying in the region under con-
sideration). The magnitude of the difference between our values and
those of de Broglie is just about the correction that would have to be
applied, if the crystal face lay in the axis of rotation of the spectrometer.
This correction has been fully explained in the article by Blake and
Duane referred to. The increase in the difference between the two sets
of values with increasing frequencies of the X-rays is what one would
expect, for the high frequency X-rays penetrate further into the crystal
than the low frequency ones, and therefore a larger correction must be
made for them.

The relation between the square root of the frequency of the char-
acteristic rays and the atomic number is approximately a straight
line relation. Moseley's original measurements of the a and'jS wave-
lengths indicate thi&. As stated above, however, they also show a de-
parture from the straight line relation in a certain r^on. This departure
occurs exactly where one would expect it to, if no correction were made
for the penetration of the rays into the crystal.

Our values of the critical absorption frequencies, which are the highest
frequencies known to be characteristic of the elements (higher even than
the frequencies of the a and fi lines measured by Moseley) fall more
nearly on a straight line than Moseley's values do. It is possible to
draw a straight line near the points representing our data such that no
point will lie as far from it as one fifth of one per cent, of its ordinate.
There is, however, a systematic variation of the points from this straight
line, which indicates that our values really lie on a line that is very
slightly curved.

The fact that after using our method of automatic correction for the
penetration of the rays into the crystal we get points that lie so nearly
on a straight line raises the question as to whether there may not be
some other correction, which we have not thought of, and which, if
applied, would make the line perfectly straight.

As is well known equations can be written out containing Rydberg's
fundamental frequency as a coefficient, that approximately represent
the frequencies of the characteristic lines of the elements as functions of
the atomic numbers. It is interesting to note (see next-to-last column
of the table) that the equation

y = y^{N - 3.5)2

in which vq is the Rydberg fundamental frequency (namely 109,675

704 ^. C. BLAKE AND WILUAM DUANE. [

multiplied by the velocity of light), represents with considerable accuracy
our experimental results. In this equation N stands for the atomic
number of the element, and the only constant determined by the X-ray-
experiments is that in the parenthesis, namely 3.5. It is interesting, too»
to note that this equation gives us the correct value for the nuclear charge
as worked out by Sanford^ from the assumption of equality between
orbital and vibration frequencies. For we have

P = — ^ (N - 3.5)*
and Qt the nuclear charge, fulfills the equation

T*e* T*A»*

Eliminating v we get at once Q = 2e{N — 3.5), an equation that mani-
festly holds to the accuracy shown in the table for the elements there
shown. Thus it would appear that a knowledge of the position of the
X-ray absorption lines, which corresponds to the head of the emission
line series leads to results that are more fundamental than a knowledge of
the position of the principal emission lines can.

It is hoped to extend the measurement to elements of higher and lower
atomic number than those included in this paper.

Harvard Unxvsrsity.
1 Physical Rbvisw, May, 19x7, p. 383*

JSg-^-] FERROMAGNETIC SUBSTANCES. 705

ON A MOLECULAR THEORY OF FERROMAGNETIC

SUBSTANCES.

Bt K6TARd Honda and Junz6 Okubo.

Index to Paragraphs.

1. Historical.

2. Calculation of internal force due to a group of elementary magnets.

3. Magnetization of a single complex.

4. Deduction of the curve of magnetization.

5. Residual magnetism and hjrsteresis phenomenon.

6. Calculation of the hjrsteresis-loss by magnetization.

7. Effect of temperature on magnetization.

8. Molecular field introduced by Prof. P. Weiss.

§ I. Historical.

According to the Amp^- Weber theory of magnetism, the molecules
of a ferromagnetic substance are all small magnets, the axes of which
in an unmagnetized state, are turned uniformly in all directions, so that
as a whole no magnetic polarity is observed. These magnetic molecules
are believed to exert directive force upon one another. If an external
force acts on the substance, the molecules tend to turn their axes in the
direction of the field in opposition to the mutual directive force. With
the increase of the field, the axes of the magnetic molecules are turned
more and more in the direction of the field ; if all the molecules are turned
in this direction, magnetic saturation is reached.

The theory was afterward improved by Sir J. A. Ewing in a most
satisfactory manner by taking into account the magnetic force due to
the neighboring molecules.^ He assumed that molecular magnets in
every microscopic crystal are arranged in a cubic space-lattice, corre-
sponding to the crystalline system of iron, which is the regular system.
In each minute crystal, all magnets naturally assume one of three
orientations of stable equilibrium, which are parallel to the sides of the
space-lattice; but as the directions of the axes of these crystals are dis-
tributed uniformly in all directions, their external action is as a whole
zero. If an external field acts on the substance, all the elementary
magnets in each crystal will tend, as a whole, to turn with their axes in
the direction of the field, but they are partially prevented from doing so

' Phil. Mag. (5), 30 (1891), 205. See also Magnetic Induction in Iron and Other Metals.

706 KdTARd HONDA AND JUNZd dKUBO. [sSam

by action of the mutual force, tending to draw these magnets back to
their original stable orientation. With the increase of field, the molecules
will more and more turn in the direction of the field and consequently
the intensity of magnetization becomes greater, tending to an asymptotic
value. Though the theory is very simple in its content, it explains
many observed facts quite satisfactorily, at least qualitatively.

R. Gans^ tried to treat Ewing's model of molecular magnets mathe-
matically; but his theory differs essentially from that of Ewing in
assuming the distribution of the molecular magnets in the substance to
be quite arbitrary and in considering the magnetic action of its neighbors
on each molecule to be constant. In Ewing's model, the mutual action
is not a constant, but a function of the angle of the rotation of molecules.
In fact the conclusions from his theory are only a rough approximation
to the observed facts.

On the base of Langevin's theory of magnetism for paramagnetic gas,
Prof. P. Weiss developed a theory of ferromagnetism,* by introducing an
assumption that every molecule of the ferromagnetic substance, though
it is not acted on by any external field, undergoes the action of a uniform
molecular field of an enormous strength amounting to several ten millions
of gauss. It is however very difficult to conceive the origin of such a
molecular field and also to explain the fundamental phenomenon r^^ard-
ing the induced magnetism by means of his theory. As is well known,
ferromagnetic elements can easily be magnetized with a field of lOO
gauss to a value of seventy or eighty per cent, of its saturation value.
If a molecular field of such an enormous strength really acts on each
molecule, how is such an easy magnetization of the substance in any
direction possible? Hence it seems now to the present writers very
probable that the existence of the molecular field as conceived by Weiss
does not correspond with the facts.

In what follows, we shall treat mathematically Ewing's theory of
magnetism exactly in the same form as put forward by himself and show
how the conclusions arrived at agree with the facts actually observed.

§ 2. Calculations of Internal Force Due to a Group of

Elementary Magnets.

According to Ewing's model, it is assumed that in every minute
crystal, or "elementary complex" as we shall call it, composing a mass
of iron, elementary magnets are all arranged in a space-lattice consisting

>Gdtt. Nachr. (zpio), 197; (1911). 118. R. Cans a. P. Hertz, Zeitsch. far Math. u.
Phys., 61 (1913). 13-

* Arch, des Sci., No. 6, 31 (19x1). 401.

XS"6^'] FERROMAGNETIC SUBSTANCES, JOJ

of squares, but that the axes of these elementary crystals are distributed
quite arbitrarily in all directions. If no external force acts on the sub-
stance, the axes of the elementary magnets in each complex take positions
of stable equilibrium, that is, towards either side of the space-lattice.
If an external force acts on the substance, the elementary magnets in
each complex are assumed to turn, as a whole, in the direction of the
field against the mutual force. The magnetization of the mass of iron
is then the sum of the magnetizations of all these complexes in the
direction of the magnetizing field. In order therefore to find the intensity
of magnetization, it is first of all necessary to deduce the law of magne-
tization for each complex.

Suppose we have a group of elementary magnets arranged in the space-
lattice and with their magnetic axes all parallel to one of the orientations
of stable equilibrium and an external field acts in the -

plane of the lattice, as shown in the annexed figure, y^ V^ y^
The action on each of these magnets by its neighbors is J^ ^ fp/rJl^
then the sum of their magnetic actions, but we may with jf"^''/r^]
a fair approximation suppose that the actions of eight L^ i/f^ ^ V^
only of the surrounding magnets are effective and those of / "^^ ^
the rest negligible. On this supposition it is easy to cal- p- j
culate the magnetic force acting on one of these magnets.

Let 2a be the sides of the space lattice, 2r and m be the length and the
pole strength of the elementary magnets respectively. We take one
side of the space-lattice as the axis of y and the other side as that of x,
the initial direction of the elementary magnets being supposed to coincide
with the direction of the y axis.

A pole of each magnet is acted on by 1 6 poles of the neighboring
magnets, and, the action of four pairs of poles neutralizing each other
by symmetry, there remain only the following eight forces:

Forces between E and P, / and P, F and P, 0 and P, Q and P, N

and P, D and P, / and P.

Now

EP^ = 4(a* + r^ - 2ar cos 6),

JP^ = 4(a2 + f * — 2ar sin B),

FP^ = 4{2d« + r* - 2ar(cos e + sxnS)],

OF^ = 4(a2 + r^ + 2ar cos 6),

QP^ = 4{2a2 + f^ + 2ar(cos 6 - sin 6)],

NP^ = 4{2a« + r* + 2ar(cos S + sind)],

IP^ = 4(a» + r* + 2ar sin 6),

DP^ = 4 {2a* + r* — 2af(cos ^ — sin d)}.

7o8

KdTASA HONDA AND JUNZd 6KUB0.

r^OOMD

ISbsbs.

Among the eight forces, those tending to increase 0 are:

JP^ 4(a« + f» - 2ar sin ff) '

OP* 4(a2 + f2 + 2af cos 6) '

nt^

QP* 4{2a^ + f* + 2af (cos 6 -sinS)]'

FP* 4{2a» + r« -. 2ar(cos ^ + sin d) } '
those tending to decrease B are:

£i» 4(a« + f» - 2ar cos B) '

IP* 4(a« + r* + 2ar sin B) '

fW^

Z>P« 4{2a* + f* - 2ar(cos d - sin ^)) '

fW'

NP* 4{2a* + r* + 2ar(cos ^ + sin d) } *

Denoting by X and F the sum of the components of
these forces in the directions of x and y respectively, we
e,^i^^^ have for equilibrium

l^x fnH sin {a — B) = — F sin d + Jf cos ^.

If we calculate X and Y from the eight forces above
Fig. 2. given and put in the last equation, we get

H sin (a — d) =

k COS B

* , * COS g ~ sin B tT I / ^ • • ^MJ

(i + 2k*)* (I — g*(cos B + sm B)*}* *^ ' ^^ ^^

r / /» . • /^Mii • ^ cosg + sin B

— [l — g(cos B + sm B)]'\ + 7 ; {rr* -j -rz t—. — rrrri

^ ^^ ^^ ' (i + 2k*)^ [i — 2*(cos dsm B)*}^

{[i + 2(cos d — sin B)]^ — [i — 2(cos B — sin B)]^] J ,

(I)

JJS"^] FERROMAGNETIC SUBSTANCES. 709

where k = a/r, p = 2Jfe/(i + jfe*) and q = 2/fe/(i + 2**). Since a > r,
.'. o < p < I and o < s < 2. Expanding the right-hand member of
the above equation in powers of p and q, we get

H (sin (a - 0)

g* P _l9 g* , 9 11 il , \_9 H
(i + 2*»)»»\3!''"2 5!''"2' 2 'r!"*" " / 2*2

r_^!__/l . 113^ , \. ^_

14(1 + *»)« V;! "^ 2 2 9! ' (I + 2ife*)*

(7!+2-t|+-)]«-*^''+-

The right-hand side of the above equation is a function of 6 only,
provided r, w, a are given. Let us denote it by F{0), If we put
F{e) = (m/r*)f(e), f{e) contains * only as a parameter. F{e) or /(^) is
evidently a periodic function of $, having t/2 as its period.

The fact, that the internal restoring force F{$) has a period of t/2
follows at once from the following physical considerations: If all the
magnets in a complex, starting from a given orientation, turn through a
right angle, the mutual action between the molecules must remain
unchanged on account of the property of the square space-lattice, and
hence F{d) must be a periodic function of t/2. In an orientation of
stable equilibrium of these magnets, there is no deflecting force acting
on any magnet due to the surrounding ones, that is, F{d) = o for ^ = o.
As the magnets deflect from this position, F{d) increases. It is evident
that for 0 = t/4, F{$) must again vanish through the symmetrical
orientation of molecules. Hence as $ increases from o to t/4, F{$) must
pass through a maximum. From $ + t/4 upward, the axes of the mag-
nets tend to place themselves in the next orientation of stable equilibrium,
that is, in the orientation for d = t/2. Hence F{d) changes its sign in
passing through T/4. As $ increases from t/4 to t/2, F{d), which is now
negative, at first decreases, attains a minimum and then increases,
vanishing at d = t/2. The same change of F{d) is repeated in the other
quadrants.

It is also to be remarked that if r be very small in comparison with a,
that is, powers of p and q higher than the third are negligibly small,
F{6) vanish for all values of d, that is, no resisting couple acts, if the
axes of the magnets be deflected from their orientations of stable equilib-
rium.

710 KdTARd HONDA AND JUNZd OKUBO, [

In the above calculation, we have only taken account of the mutual
force due to eight surrounding magnets. If we take the next i6, 24,
32, ... magnets in the outer squares into consideration step by step,
the expression for F{6) rapidy converges to a definite value; because
for each outer square, the number of magnets increases by 8, while the
force exerted by each pole in different squares, decreases by the inverse
square of the distance. For example, if 24 magnets in the first two
squares be taken, the amplitude in F{d), assuming jfe = 2, increases only
by 3.6 per cent, as compared with the case above discussed. Moreover,
by taking all the magnets in the complex into consideration, the period-
icity of F{0) cannot evidently vary for the reason as explained above.
The only change consists in the variation of the coefficients of sin 46
and sin* 2d. More generally, if we consider the distribution of magnets
in a cubical space-lattice and the effect of the magnets situated in two
adjacent planes on the magnet under consideration, it is easily found
by calculation that the correction due to this effect amounts only to
4.8 per cent, as compared with the case before mentioned. Hence we
can conclude that in the most general case, F(0) is a periodic function
of 6, having v/2 as its period and jfe as a parameter.

The expression for F{d) may generally be written as

F{e) = A sin 4^,
where

and <p, <p' are the functions of k only, k being always greater than i.
For Jfe = I, the amplitude of F(d) is infinitely large; as k increases from
I, the amplitude rapidly decreases, the value of tp' becomes very small
in comparison with that of <p, and the form of the curve approaches to
F(e) the sine as given by the first term of the above series.

Fig. 3 shows this manner of approaching the sine
curve; here curves i, 2, 3 are those corresponding to
fe = 1.3, 1.6, 2.0 and curve 4 represented by a broken
line is a sine curve. Their amplitudes

Ai :At :Ai = 4431 : 0.976 : 0.256

are all reduced to the same magnitude as that of the
sine for the sake of comparison. Thus we see that for
a value of k greater than 2, the form of the curve F(d) is very nearly
equal to sin 46. In the case of iron, nickel and cobalt, which are easily
magnetizable, this restriction seems to be quite reasonable. Hence,
under this limitation, we may use, for the first approximation, A sin

J52J"^] FERROMAGNETIC SUBSTANCES. 7 1 I

4$ with a constant amplitude instead of F{6), and proceed to develop
the theory of magnetization. Relation (I) takes then the following form

H sin (a — 6) ^ A sin 4$ . (2)

and

•

where ^ is a function of k only. As shown in Fig.
4, <p rapidly decreases with increasing k.

If a special investigation be necessary for the case ^
of a closer molecular distance, we must use the
exact relation (i). But, as we shall see presently, ^^
we have always used a graphical solution for rela-
tion (2) and consequently the substitution of rela-
tion (i) for the last one does not cause much com-
plication in our calculations.

6 K

Fig. 4.

§ 3. Magnetization of a Single Complex.

Suppose in a complex an external field H acts in a plane parallel
to the face of the elementary cube and in a direction making an angle a
with one of the directions of stable equilibrium of the molecular magnets
arranged in the space-lattice; the magnets will then be in equilibrium
after turning through a common angle 0 from their initial direction.
It is required to find the component of magnetization / in the direction
of the applied field. We have obviously

where n is the number of elementary magnets and /© the saturation value
of the intensity of magnetization. Denoting //Jo = i, we have from
the above relation

i = cos (a - 6), (3)

in which a, 0, H are related by an equation

H sin (a — e) -A sin 4^;

if we denote HI A = A, we get

h sin (a — ^) = sin 4^. (4)

A contains w, r, a and depends on the properties of particular substance;
so also /o. But if we use the reduced i and h instead of the actual
intensity of magnetization and field, relations (3) and (4) apply for all
ferromagnetic substances belonging to the regular system. If h and a
be given, equation (4) gives the value of 6 and therefore equation (3)

712

k6tAR6 HONDA AND JUNZd 6KUB0.

the value of i. Hence equations (3) and (4) may be considered as the
laws of magnetization.

Since equation (4) does not change, if we put for a and 6 the values
a + t/2 and $ + t/2, it follows that the force required acting in the
direction a to deflect the system of magnets through an angle 6 is equal
to that acting in a direction a + t/2 and deflecting these magnets by
$ + t/2. Hence the curve of magnetization by a force acting in a direc-
tion a partiy coincides with the curve corresponding to a system of
magnets, whose initial direction makes with the field an angle a + t/2.

If h and a be given, 6 can be found from equation (4), which is of
the eighth degree in sin ^ or cos 6; hence we can not solve it analytically.
However, as ^ is given as the intersections of the two curves

y = sin 48 and y = A sin (a — d),

we can easily find it by a graphical method. In Fig. 5, curve / represents
y = sin 4^, and curves a, 6, c, d, those of y = A sin (a — $) for a = 30®,

Fig. 5.

Fig. 6.

70**, 120** and 160® respectively, h being taken as 0.6. By giving different
values to A, the curve of magnetization can be obtained.

In Fig. 6, four curves representing the relation between i and h are
given, in which for the angle a were taken angles of 30®, 70®, 120® and
170** respectively. They give the intensity of magnetization in the direc-
tion of the respective field, when the magnitude of the latter is so varied
that it is always in equilibrium with the internal resisting force sin 46,
In the curve for a = 30®, the initial point a corresponds to the value of
cos 30**; as k increases, $ becomes greater, but always less than a, and
therefore i — cos (30® — 6) steadily increases, tending asymptotically
to the value of i = i with A = 00 . In the curve for a = 70**, the point
b corresponds to the value of cos 70**; as A increases from o, $ and there-
fore sin 4$ also increases. Since however the latter quantity reaches a
maximum at ^ = t/8, A must be diminished from a certain value of 6
upward, if the magnetization is to be effected statically or reversibly.
With $ = t/4, the resisting force sin 4$ vanishes and therefore A must be
diminished to zero; with a further increase of d, sin 4$ changes sign and

No. 6. J

FERROMAGNETIC SUBSTANCES,

7^3

therefore h must be applied in an opposite direction, if the magnetization
is to be made reversibly. If $ approaches to 70**, h becomes — 00 in
the limit and the magnetization tends asymptotically to unity. The
curve for a = 120®, which begins at the point c on the negative side of i
passes through a maximum and a minimum of h, and coincides with the
curve for a = 30**, as the value of i increases. The curve for a = 170®,
b^^ning at a point d on the negative side of i, passes through two
maxima and one minimum of h with the increase of i and approaches
asymptotically to the line i = i.

In the ordinary case of magnetization, the field is continuously in-
creased, and therefore the magnetization is only partly reversible. But
it is easy to see in what manner the magnetization in the direction of
the field increases by applying a continuously increasing field.

Case I , o < a < (t/4) . The component magnetization i in the direc-
tion of the field increases with h and becomes i for A = 00 . If the field
is gradually reduced, i takes its original value, and there is no hysteresis.

Case 2, (t/4) < a < (t/2). i increases with h continuously up to the
maximum resisting force; here it undergoes an abrupt change and takes

**-'"

* %

Fig, 7.

Fig. 8.

Fig. 9.

Fig. 10.

a value corresponding to the rotation of t/2 of the initial orientation of
molecular magnets. With a further increase of the field, i continuously
increases in a manner, as if the initial orientation were a — (ir/2). If
the field is reduced, i takes a value quite different from its initial, as
shown in Fig. 7; that is, there gives rise a hysteresis phenomenon.

Case 3, (t/2) < a (3/4)t. i increases with A, at first continuously, and
then abruptly, when the resisting force reaches a maximum. After
this, the curve of magnetization follows the course corresponding to the
case with the initial orientation of a — (t/2) (Fig. 8). With the reduc-
tion of the field, hysteresis phenomenon is also observed.

Case 4, (3/4) T < a < T. The curve of initial magnetization is the same
as in the above cases. If the first maximum of the resisting force is
less than the second maximum, its next magnetization is the same as
in the case with the initial orientation of a — (t/2) (Fig. 9); if the
first maximum is greater than the second, the magnetization is the
same as that for the initial orientation of a — t (Fig. 10).* The subse-

714

KOtAR6 HONDA AND JUNZd dKUBO.

quent magnetization takes place continuously. By reducing the field,
the corresponding hysteresis is observed.

The relation between the initial orientation and the maximum resisting
force hm can be found in the following way:

From

sin 4O

we have

sin (a — d) *
dh ^ 5 sin (g + 3^) + 3 sin (a - 5^)

2 sin^ (a — e)

i.0
1.0
1.0
1.0

I , >

*-^a

to' 40° 60° 80° 100' U0° UO ISO* IN

Fig. 11.

If the value of d corresponding to the maximum force be denoted by Oq
we have

5 sin (a + 3^0) = 3 sin (5^ - a) (5)

and

_ sin 4^0

* sin (a — ^0) '

The existence of such values of do can be understood from Fig. 6. The
calculated values of h^ for different values of a are given in the following
table and in Fig. 11.

«.

*-.

«.

A..

«.

A«.

«.

>^.

45**
50**
60**

4.000
2.625
1.750

100**
120**
140**

1.025
1.008
1.137

70**
80**
90**

1.405
1.205
1.088

160**
170**
180**

1.541
2.018
4.000

Curve a in Fig. 1 1 refers to the first maximum ; in the interval between
135® and 180®, the second maximum is also possible. However, as h^
corresponding to a for the first maximum is equal to that corresponding
to a + (ir/2) for the second maximum, curve b for the second maximum
has the same form as curve a being only displaced through t/2.

§4. Deduction of the Curve of Magnetization.

Hitherto we have exclusively considered the magnetization of a single
complex; but we are now able to study the magnetization of a mass of

flS^^] FERROMAGNETIC SUBSTANCES. 715

ferromagnetic substance, such as iron, which consists of a great number
of such elementary complexes with their magnetic axes uniformly dis-
tributed in all directions. Now, the faces of the elementary cubes or
the complexes are in actual cases directed uniformly in all directions;
but for the sake of the simplicity of calculation, it is here assumed that
the complexes have one of their faces all parallel to a common plane,
other faces being distributed quite arbitrarily, and the magnetic field
acts parallel to this plane. The problem is then reduced to the two-
dimensional. The magnetization of this simple case does not obviously
differ from that of the actual case in its character.

Let N be the number of elementary complexes; if there is no magnetic
force acting on these complexes, the number of complexes, whose mag-
netic axes make, with a certain direction, an angLe lying between a and

a + da, is equal to

N
dN = — da.

If Af be the magnetic moment of a complex, whose magnetic axis makes
initially an angle a with the direction of the field, then the component
of magnetization in the direction of the field is Af cos (a — ^). Con-
sidering JIf to be the same for all complexes, the total magnetization
due to these complexes is

/ = I cos (a — e)dd = — I cos (a — e)da,

t/_, 2ir T Jq

i = - I cos (a — B)da, (6)

The relation connecting a and B must however be different from that
for a single complex. Here besides Fifi)^ we must also consider the
magnetic force due to surrounding complexes. If no field acts on the
substance, the resultant effect of the surrounding complexes is obviously
zero; but in its magnetized state, this is not the case. To calculate
this force exactly is almost impossible; but it is not difficult to calculate
approximately its mean effect. Since the total action of a complex
on a magnet within it is the same as the sum of the effects of neighboring
magnets, those of the distant ones being very small, we may consider
the form of the complex under consideration to be a sphere, without
causing sensible error in the value of F{B). The magnetic effect of
other complexes on the magnet under consideration may approximately
be replaced by that due to a uniform distribution of magnetization with

7l6 KdTARd HONDA AND JUNZO 6kUB0,

a mean intensity in the space in which other complexes are found. As
the boundary of the said complex is assumed to be a sphere, this force is
(4/3)7/ acting in the direction of the external field and does not generally
coincide in direction with that of the axis of the magnet under con-
sideration; and hence it exerts a couple tending to turn the magnet in
the direction of the field. Hence instead of relation (2), we must use
the following formula:

yH + -J tI sin {a " e) ^ A sin 4^.

But for a given value of -ff, / is a constant, so that for a while we may
regard H + {^li^irl as an external field and proceed to calculate / for
different assigned values of ff + {j^I^tI, After finding /, the actual field
may be found by simply subtracting (j^irl^I from the assigned field.
Hence the sanxe relation as (2) may also be used in the present case,
that is,

- sin 4^

sin {a — B)'

ii)

If A be given, equation (7) gives B in terms of a, and if this value of
B be substituted in equation (6), this gives the intensity of magnetization
i in terms of A, and thus the problem is formally solved. But in actual
calculation, some complications are involved, and we must separately
consider cases corresponding to several graded values of h.

First let us consider the case when h is very small ; then B is also small,
and therefore sin 4^ = 4^. From (7), we get

h (sin a — B cos a) = 4^;

h sin a

B =

4 + A cos a '

Equation (6) gives

' ^ r r . z. • N^ I r J ^ ft sin« a \ .

t = - I (cos a + ^ sm a)da « - I \ cos a H ;— 7 ) da

tJo it Jo I ' 4 + AcosaJ

= I cos aoaH I sm' a- 1 I +-cos a| da,

frJo 4irJo ^ 4 ^

The first integral vanishes; and if the second term be expanded and
intergrated, we have

A /I ¥ T h* \

= 0.125A + 0.00195A' + 0.00007A* + • • •. (8)

JJJJ"^^] FERROMAGNETIC SUBSTANCES. JlJ

As it ought to be, i is an odd function of h. If A be sufficiently small,
the terms of any higher order of h than the third can be neglected, and
i and h are linearly related to each other. This fact was verified by
experiments of Bauer,^ Lord Rayleigh* and others. In this case, the
magnetization is perfectly reversible, that is, there is no hysteresis, a
fact which agrees with the result of the experiments.

Secondly, we consider the case, where h is large. To change the
integration variable from a to $, we differentiate equation (7),

A cos (a — ^) I ^ — I j = 4 cos 4^

And also

da ^ 4 cos4g

d^ "* Acos(a — ^) "^ ^'

cos (a — ^) = ± r ^A* — sin* 4^

(9)

According to the magnitude of A, all the complexes, during magnetiza-
tion, do not necessarily change their angle of deflection continuously;
in fact, some of these complexes made an abrupt rotation of t/2 or x.
Hence in evaluating the above integral, it is necessary to divide the
limits of integration into several parts. If A be given, we can find
from Fig. 1 1 the value of a having A as A,,,; the values of $ for these values
of a may then be found from equation (7). We have generally three
values of a and 6, let us call them by ai, as, at and 61, 62, 9%> Then we
have

•/O •/ai *^a% *^m%

In the first and fourth integrals, the elementary magnets in the com-
plexes belonging to these integrals remain stable, since the field is less
in their C£ises than the critical value. The magnets in the complexes
belonging to the second integral all lie beyond the position of stable
equilibrium, and therefore the magnetization is the same, as if the initial
orientation of the complexes were a — (t/2). Hence the limit of the
second integral must be changed from ai and at to ai — (t/2) and
«i — (x/2). In the third integral, the magnets in the complexes lie
beyond the first and second positions of stable equilibrium, and therefore
the magnetization is the same, as if the initial orientation were 2 — x.

^ Bauer, Inaug. Diss. ZOrich (1879). Wied. Ann., II. (1880). 399.

* Phil. Mag., March (1887). See also Ewing's " Magnetic Induction," 134.

7i8

KdTARd HONDA AND JUNZO OKUBO.

LSbsxbs.

Hence the limits of the third integral are to be changed from a% and at
to ai — T and a% — t. If the integration variable be then changed
from a to ^, we have

'0 •/#i •J$t

Now from equation (9), we have

Jo Jit •/#• •/#. J»A

i = ^|sin4^|r±^J J I -^sin2 4^(f^

-7 (sin 4^ — sin 4^0) ± - j i ^ i — *' sin* 4$ dS

-r-^

jfe2 sin* 4^^

!■

where fe' = i/A*. Hence if £ be an elliptic integral of the second kind,
we have

t = -^ (sin 4^ - sin 4^) ± — {£(*, 4O - £(*. 4^)}. (10)

ir/^ 4^

By expanding £ in a power series of k, we have

i = ;jjr(sin4^ -sin 4^) ±~;:|{4<^- ( j,4^ + ^sin 8(?) **

- ( ^!- •4<^ + —-(— + - sin*4(?) sin ^e \ k^
[ 2*4*3 242 \24 4 ^ /

-(ft'^!--4<^ + ^-^(z^^ +T^sin*4^ + 2sin*4^)-sin84*' (ii)
(2*-4*'6* 5 ^ 2-4'6\6-4-2 64 ^6 ^ /2 J , ^

The double sign of the second term must be so chosen that upper and
lower signs correspond to a — ^ > (t/2) and a — d < (t/2) respectively,
with the condition that if an abrupt turning of the molecules through ir/2
takes place, a and d are measured from the new position of equilibrium.

In the following tables and in Fig. 12, the result of our calculation
according to the above relations is given. Up to A = 0.5, i was calculated
by relation (8), while for higher fields, it was obtained by means of
relation (11). Thus, we found at first three values of a corresponding to
different values of h:

«i.

««.

Of.

«i.

«t-

«••

1.5
2.0

tT 0'
54'' 38'

157** 0'
145'* 0'

159'' 30'
169" 0'

2.5
3.0

50° 30'
47" 0'

140" p'
138" 30'

174" 0'
178" 0'

Na6. J

FERROMAGNETIC SUBSTANCES.

719

From equation (7) and these values of a, we found the following values
of 6 for the limits of integrations:

9i.

0i'.

9%,

»•.

fc'.

•4.

1.5

0""

-8**

21*

- 8*

-6*

16*

2.0

26«

-13*

26*

-13*

-4*

14*

2.5

O**

34''

- 17* 20'

34*

- 17* 20'

- 2* 30'

12* 30'

0°

3.0

0«

37**

- 21* 30'

37*

- 21* 30'

- 1* 30'

We have then for each of the integrals the following numbers:

^'-

Sum.

1.0

• • • •

• • * •

0.183

1.5

0.301

0.436

0.036

-0.096

0.677

2.0

0.286

0.448

0.127

-0.045

0.816

2.5

0.289

0.475

0.172

- 0.031

0.875

3.0

0.241

0.483

0.196

- 0.011

0.909

Thus the form of the curve of magnetization agrees precisely with that
experimentally found. This curve starts from the origin at a definite
angle, and increases at first linearly with the field. With a further
increase of field, the magnetization increases more and more rapidly;
in a certain field, its rate attains a maximum
and then gradually decreases. The curve
of magnetization passes therefore through
an inflexion point, and gradually approaches
to an asymptotic value i, as the field is
increased. The curve is the normal curve
of magnetization with the reduced intensity
of magnetization and field; it is common
for all the ferromagnetic substances belong-
ing to the regular system. The curve

of magnetization for a particular substance can be obtained by multi-
plying Iq and A, characteristic constants for the substance, to io and h
respectively.

If the curve of magnetization be plotted against the actual field as
explained at the beginning of the present paragraph, the characteristic
form of the curve will not materially change.

§ 5. Residual Magnetism and Hysteresis Phenomenon.

If a mass of iron is once magnetized to saturation, and then the field
reduced to zero, there remains a residual magnetism. The amount of
this residual ms^netism can easily be found in the following way: The

Fig. 12.

720 KdTARd HONDA AND JUNZO 6KUB0. l^S

complexes, whose magnetic directions lie initially between o and t/4
will return to their original position with A = o; the complexes, whose
magnetic directions were initially ir/4 > a > t/2, or ir/2 > a > (3/4)x,
take a new position of equilibrium differing from the initial by t/2
with A = o. Lastly the complexes, whose magnetic directions were
initially (3/4) t > a > t, will come to a new position differing by x from
the initial with A = o. Hence, if the field be reduced to zero, the mag-
netic directions of all the complexes are distributed uniformly within
an angle making 7/4 on both sides of the field. The residual magnetism
may therefore be found thus:

nwl4

i? = 2 I Af COS BdN. dN -- — de

4/0 r^ , 4^1

^ ' cos W^ = ^

^ Jo ir^2 '

Hence the reduced residual magnetism r is

r = 7- = 0.8927. (12)

This is the same value as obtained by Ewing.^ Thus there remains a
residual magnetism of about 90 per cent. The experiments with very
long iron wires confirm the correctness of this conclusion.

According to the above consideration, the process of reducing the
field from 00 to o is reversible, that is, the magnetization during the
reduction of the field from 00 to o exactly coincides with the magnetiza-
tion from A = o to 00 , the initial magnetization being r . This curve of
magnetization can easily be found : because the initial orientation of the
complexes is known to be uniformly distributed within an angle sub-
tended by the lines inclined at t/4 to the field. If A be small,

4 r'* , X . , , sin 4^

i = - I COs(a — djda and A = -: — ;; rr ,

T Jo sm {a — e)

I /"'* A

.*. * = - I (A + 4 cos a)(l + - cos a)''^da
xJo 4

= 0.8927 + 0.047A — 0.083A* + • • • . (13)

For a large value of A, we find from equation (7) the value of d corre-
sponding to the limit of integration. By means of equation (11), the
value of i will be found on simple substitutions.

Starting from the residual magnetism, the magnetization by a
gradually increasing negative field can be calculated in a similar way.

* Magnetic Induction (1900), 325.

FERROMAGNETIC SUBSTANCES.

731

This case is equivalent to the magnetization by a positive field of a
group of complexes, whose initial magnetic directions are uniform and
given by ± (3/4)^ > a > t. For small values of h, we have

i= -^ {' cos{a-e)da= -- r (A + 4cosa)(i +-co3o)-ya

"■»/«/«• 'Jv*w 4

= + 0.8927 — o.04,7A - 0.083A' — • ■ ■. (14)

For large values of h, we find i from equations (7) and (11), as in the
former case. The results of calculation are included in the following
table:

,-.

+ 00

1.000

- 1.0

0.815

2.0

0.944

-3.0

-0 847

3.S

0.973

-1.5

0.015

1.S

0.932

-5,0

- 0.981

3.0

0.962

-2.0

-0.584

1.0

0.922

— 30

-1.000

2.5

0.956

-2.5

- 0.786

0.0

0.893

In this way, we can obtain a well-known hysteresis loop, when the field
is varied between + "> and — «, as shown in Fig. 13. It possesses all
the characteristics shown by iron, nickel and cobalt, and is far nearer
the experimental curve than the rectangular hysteresis loop obtained
by Gans.

The hysteresis loop accompanying a cyclic change of magnetic field
between -f A and — h can be calculated in a simi- ^

lar manner. For this purpose, the residual magne-
tism obtained by reducing the field from A to 0
will be at first calculated. Then, the curve of
magnetization having this residual magnetism as
the initial will be calculated, it must coincide with
the curve of demagnetization obtained by reduc-
ing the field from h to o. Next, the curve of '
magnetization from o to — A, having the state of re-
sidual magnetism as the initial, will be calculated,
and so on. In this way, we have obtained a complete cycle of magnet-
ization.

The residual magnetism, when the field h is reduced to zero, is easily
known; because for a given value of h, we can find from Fig. 11 the
values of a having h as the maximum resisting force, and therefore it
can be completely known how many complexes, which had initially a
unifonn distribution of their axes, will return to their original position
by reducing the field to zero and how many of them will rotate through
one or two right angles from their initial positions. Hence the residual
magnetism can be calculated by the following expression:

Fig. 13-

2 KdTARii HONDA AND JUNZd dKUBO. [ISiS

C03 (a — T)da + I cos ada \

Since the orientation of the m^;netic axes of these complexes in the
residual state of magnetization is thus completely known, a further
magnetization with positive and negative fields can be calculated in
the same way as tbe case above discussed. In this way, we calculated
three curves of hysteresis for different values of A, which are shown
graphically in Fig. 14. The curves are found to agree with the results
of experiments.

In our theory, the hysteresis phenomenon takes place only when the
molecular magnets in the complexes turn abruptly; otherwise the process
of magnetization should be reversible. Thus, as we have seen, the initial
magnetization up to about h = i and also the demagnetization and the
second magnetization between o and A, ought to be reversible. In
actual cases, however, we also find a small but distinct hysteresis in
weak fields. This discrepancy between theory and experiment may
probably be due to two causes, which are not considered in the above
theory.

In the important paper' on the modulus of rigidity of rocks, Prof, S,

Fig. 14. Fig. 15.

Kuskab^ has shown that by cyclically changing the twist between
+ T and — T, all rocks investigated by him show a distinct hysteresis,
though Hooke's law is fairly well satisfied. The form of his hysteresis
loop is quite similar to that observed in iron in weak fields. As an
example, we reproduce here his hysteresis curve of twist for marble
(Fig. 15). He explained the phenomenon quite satisfactorily by his
theory based on the experimental fact that by applying couple, the twist
of the specimen, after its instantaneous increase of a definite amount,
gradually increases with time, asymptotically tending to its final value,
that is, the twist shows a time-effect.
> Joum. Coll. Sci.. 19. Art. 6 <I903}.

f%^^] FERROMAGNETIC SUBSTANCES. 723

Now the magnetization has also a time-effect called the magnetic
viscosity, though it is not so conspicuous as it is in the case of rcoks.
Namely, the magnetization does not instantaneously increase to its final
value by applying a magnetic field, but it requires some time for arriving
at its maximum value. This effect is specially conspicuous in weak
fields, and may therefore be considered as the first, but less important
cause of the hysteresis observable in weak fields.

The second, but principal cause of the hysteresis in weak fields is
probably the irregular distribution of the axes of the elementary magnets
situated on the bounding surfaces of different complexes. In our theory,
we have assumed that if there acts no external fields, all the elementary
magnets in each complex assume the same direction of stable equilibrium.
But in actual cases, the elementary magnets on the bounding surfaces
of the complexes may place themselves in quite different directions, as
do those in the interior through the action of the magnets in the neighbor-
ing complexes; and thus there results an irregular distribution of ele-
mentary magnets on the bounding surfaces. Hence some of the elemen-
tary magnets may initially be found in positions, which are not far from
those of unstable equilibrium. If a weak monetizing force acts on
such magnets, it may cause the abrupt rotation of the magnets and
therefore a hysteresis phenomenon results even in a weak field.

§6. Calculation of the Hysteresis-loss by Magnetization.

According to our theory, the hysteresis-loss takes place only when the
rotation of the molecular magnets caused by the external field becomes
discontinuous. That is, if the reduced field h be less than i, there is no
sudden rotation of molecules, and hence no hysteresis-loss by magnetiza-
tion ; if however h be greater than i , some of the molecules make abrupt
rotations and give rise to the hysteresis phenomenon. The number of
such molecules will increase with the strength of the field and attain to
an asymptotic value at A = 4. A further increase of magnetizing field
does not cause any more abrupt rotation of molecules.

In the curve of magnetization, O AB in Fig. 13, the hysteresis-loss
takes place only in a portion (A = i to 4) of the m^netization curve.
During the demagnetization from A = 00 to o, no abrupt rotation of
molecules occurs, and therefore we have no hysteresis. But the magne-
tization in the opposite direction from A = o to — 4 involves a loss of
energy. Similarly, in portion DE of the magnetization curve, there is no
loss of energy, but in portion £5, we have a loss of energy equal in amount
to that in portion CD.

According to the general theory of m^netism, which assumes no

724 KdTARd HONDA AND JUNZO OkUBO, [i

hypothesis as to the molecular magnets, the total loss of energy during
a complete cycle is equal to the area of the hysteresis-loop. By our
theory, the hysteresis-loss is the kinetic energy obtained by the molecules
during their abrupt rotations, and hence it is very interesting to investi-
gate, whether in a cyclic process of magnetization, the kinetic energy
thus obtained is equivalent to the area enclosed by the hysteresis-loop.
As the following calculation will show, the result completely agrees with
the above theory; moreover, in the process of magnetization, we can
distinguish the energy dissipated during the magnetization from the
total energy.

We shall at first consider the energy loss of a single complex during
m^netization. If h increases from o to A«»» which is the critical field
for the abrupt turning, the molecular magnets in the complex will turn
reversibly towards the field; at A =» A«, an abrupt turning of the mole-
cules occurs, and their axes take new orientations corresponding to the
initial position differing by t/2 or t from the original. During the abrupt
turning, the molecules will acquire a kinetic energy, which is nothing but
the heat energy produced ; the quantity of this energy must be equal to
the sum of the work done on the molecules.

The couple N acting on a molecular magnet, whose magnetic moment

is Af , is

N = M[H^ sin {a - 6) - A sin 4^},
where

M = 2fnr,

If Ba and Bi be the angles of deflection of a molecule from its initial
position, which correspond to the positions just before and after the
abrupt turning, we have

NdB = 2Jlf I [H^ sin (a - ^) - ^4 sin ^B\dB,

where the summation is to be extended to all the molecules n in the com-
plex. Since Af , a, B are the same for all the molecules, the above equation
may be written as

W ^ nM \ [H^ sin {a - B) - A sin ^B\dB,

w c^

^« = UTT = I {*- sin (a - ^) - sin ^B}dB, (i)

fiMA

where w, is the reduced hysteresis-loss by magnetization. The latter
does not involve any quantity depending on the nature of a substance;
it is therefore applicable for all substances belonging to the cubic system;

VOL.X.1

Na6. J

FERROMAGNETIC SUBSTANCES,

725

Now the reduced intensity of magnetization for a single complex is

t, = cos (a — 6) ;

for a small variation in the magnitude of A, we have

dis = sin (a — 6)d6.
Hence

w, = I hmdi — I sin ^OdO = I h^di + I sin ^ddd — I sin 4$d$.

But the last two terms are the integrals along the reversible courses of
magnetization, in which case we have the relation

h sin (a — ^) = sin 4^,

/. A sin (a — e)de =^ h di, = sin ^BdO.

Moreover, the molecular magnets at ^1, have the same potential energy
with regard to the axis ^ = o, or that perpendicular to it. Hence

sin ^ede = I sin ^BdOi = I ddi,

J 9 it •//!'

and therefore we get finally

nil nh nh

w, = I hmdi, + I hdi, — I hdi,.
Ji^ J If/ Ji\'

(2)

D
A

A-

B\e{>

Fig. 16.

Referring to the annexed figure (Fig. 16), in which ABB'C is the curve
of magnetization and B'A' the course taken by the
magnetization curve, when the field is reduced to zero,
we see that the first integral represents the area DBB'D'
and the second the area ABD and the third the area
A'B'D', so that w, is equal to the area ABB' A'.

Next, consider the case of the mass of a ferromagnetic
substance consisting of an immense number of minute
complexes, whose magnetic axes are uniformly distributed
in all directions. From the above result, we see that if
I\ and 1% be the intensities of magnetization of a complex corresponding
to the magnetizing and demagnetizing stages for the same strength of
field, we have

w. = r"(/2 - h)d}i = r (/, - ix)dK

«/o «/o

where A may take any value whatever, as w, vanishes for larger values
of A than A.,. Hence, for the hysteresis-loss w of a mass of the ferro-
magnetic substance, we must summarize the above expression for all
the complexes constituting the substance.

726 KdTARd HONDA AND JUNZO dKUBO, [&SSS

=^«'-=r

/. w = Xw. = I 2(/i - Ii)dh;

but Z/i and 2)/t are respectively the reduced intensities of magnetization
corresponding to the ascending and descending branches of the magnet-
ization curve. Hence putting

2/1 = i, and 2/2 = is,

-£(H-i^

w = J dt- ii)dh. (3)

Referring to Fig. 17 ^ w represents the area OABC enclosed by the

magnetizing and demagnetizing branches (i) and (2) of
the magnetization curve. The area OABD is known to
be the total energy of magnetization, and therefore the
area CBD^ which is the difference between the areas OBD
and OBC^ corresponds to the net energy of magnetiza-
tion.

In the same way, it can be shown that in a cyclic pro-
Fig. 17. cess of magnetization, as shown in Fig. 13, the hystere-
sis-loss during the magnetization CD is given by the area
CDE and the loss during the magnetization EB by the area BCE, and
that the total loss during the cyclic magnetization is equal to the area
enclosed by the hysteresis-loop.

In our theory, we have assumed that if no external field acts on them,
all the elementary magnets in each complex assume the same direction
of stable equilibrium. But in actual cases, the elementary magnets
on the bounding surfaces of the complexes may place themselves in
quite different directions as do those in the interior through the action
of the magnets in the neighboring complexes, and thus there results an
irregular distribution of elementary magnets on the bounding surfaces
of the complexes. Hence, some of the elementary magnets may initially
be found in portions corresponding to A = A«». If a weak magnetizing
force acts on such magnets, it may cause the abrupt rotation of the
magnets, and therefore the hysteresis phenomena result even in weak
fields. The small hysteresis usually observable in portions OA, CB and
DE in Fig. 13 are explained in this way.

I. We shall next calculate the value of the reduced hysteresis-loss for
different magnetizing fields. Now

^ = ~ I I {A« sin (a — ^) — sin ^B\dBda
'^ Jo J$o

= - 1 I A,»| cosin-— a " Bij — cos 4^0 [

(4)

nS"^] ferromagnetic substances, 727

+ - {cos 4^1 — COS 4^0} I da,

where n = i or 2 and hm, a, Oo, 61 are related by the equations of condi-
tions:

hm sin (a — ^0) = sin 4^01

A« sin I « ^ - a - Bij = sin 4ft, (5)

5 sin (3^0 + a) = 3 sin (5^0 - a).

If we eliminate from these four equations a, $0, 61 the required relation
between w and hm will be obtained; it is, however, very difficult to find
actually an analytical expression for w\ but the problem can be solved
graphically without any difficulty. Since w is the reduced hysteresis-loss
applicable for all substances crystallizing in a cubic system, it is sufficient
to find its value once for all in some convenient way; from this value,
the actual hysteresis-loss for a given substance can be obtained simply
by multiplying it by the product /o4, depending on the properties of
the substance.

The curve representing the relation between hm and a is given in Fig.
11; hence if hm be given, the corresponding value of a can be known.
If from the first and third equation of condition, a be eliminated, the
relation

^^hm^ - I

COS 460 = ±

is obtained, which gives ^0 in terms of hm- The double sign can be deter-
mined without ambiguity. Knowing hmi a, «, ^1 can be obtained from
the second equation of condition (5). Thus, from the given value of hmt
all quantities under the integral sign in expression (4) can be evaluated.
Now, from the first and third equations of conditions, we get

dct I / ^ N

^=-^-tan («-«?.);

hence for w, we obtain the expression

if I r^ da

where
<P = hm\ cos In- — a — 61) — cos (a — ^0) [+ - {cos 4ft — cos 4^0} ;

ip can be graphically evaluated, provided hm is given. Then draw the
curve and evaluate the area bounded by the curve and the abscissa;

t^J^ZJ. jL.fZ JZj

^-m.* r

-^ t:ii» ^crsin -E jx

-3C 3a*CWTEf 32.1ier

kxzs r

X _*!.:

:.*>:

f.l

m -^ -

t r- 1

I XK

9
%

TwnrTiTn'^.

to-

-J *j :a ftj

u «J

F'j{. :i-

?* :

: *»^

'5 I

4.:

* * • •

of a substance viixh has pre%~xu24y been
can be ca tenia tffl. Tbe results for ^da.^
lfJ\VjWLO% t2Lble 2tnd in F%5- 20 and 21.

IS traec^posTe

ard ar

t'«*.

1»«

!.>»

L> U U

nj.20

1* •

LJ sj r« u
Fir- 21

».

^^»

1.0

0.000

0 665

2.0

0.998
1945

2.5

3.0

2.445

2,820

i5
4.0

2.565

2-<»5

X^^] FERROMAGNETIC SUBSTANCES, 729

Here the initial increase of <p{da/dhm) and w is comparatively less
abrupt than in the former case. The double value 2w is equal to the
loss during a cyclic process of magnetization. The dotted curves in
Figs. 19 and 21 are the supposed ones, in which an irregular distribution
of the molecular magnets on the bounding surfaces of different complexes
already referred to is taken into consideration. The dotted curve in
Fig. 21 resembles in its character with the curve given by the Steinmetz
formula, that is,

where 17 is a constant depending on the nature of a substance and B
the magnetic induction.

§ 7. Effect of Temperature on Magnetization.

In the above theory, we have taken no account of the thermal motion
of the molecules, and therefore the results so far obtained hold good
only in the absolute zero, where no thermal agitations exist. In this
paragraph, we shall consider the effect of temperature on magnetization,
the established facts of which may be summarized in the following words:
In a very weak field, the magnetization increases with the rise of tem-
perature, at first slowly and then very rapidly, and after reaching a
sharp maximum, it falls very rapidly at the critical temperature. With
the increase of magnetizing field, this effect of increasing magnetization
becomes continuously less. In a field of several gausses, the magnetiza-
tion remains constant up to the critical range, and then falls very rapidly.
With further increase of field, the magnetization b^ins gradually to
decrease from a temperature which is lower as the field is stronger.
Above a field of some hundreds of gausses, the magnetization b^ns
gradually to decrease from room temperature.

It is commonly admitted that the diminution of magnetization at
high temperatures is due to the rotational vibration of molecules, the
amount of diminution increasing with the amplitude of vibration, and
that when the rotational vibration is changed into a continuous revolu-
tion, magnetization completely disappears. Such an explanation as-
sumes no change either in the molecules or in their mutual configuration ;
what is assumed is simply the change of the amplitude of the rotational
vibration during the heating. It is however questionable whether this
is sufficient to explain the so-called magnetic or At transformation.^
We shall at first show that simple revolution of molecules about their
own centers are not sufficient to account for the disappearance of magnet-
ism at the critical point.

^ K. Honda, Sci. Rep.. 4 (1915). 169.

730 K6TAR6 HONDA AND JUNZO OKUBO. [^JS

Consider the case when the external field is very strong and the
mutual action between the molecular magnets can be neglected. All
the molecular magnets are then directed nearly in the direction of the
field. Owing to their thermal energy, they make translational and
rotational vibrations about their mean positions. If 2/3 be the complete
amplitude of the rotational vibration of a molecule, its equation of
rotational motion will be

K-TT = — 2rHsinfi,
or

— = - n> sm /3, »* = -;^ » (0

where K is the moment of inertia of the molecule about the center of
mass, H the external field and 2r the pole distance of the molecule.
Suppose at first fi < t and integrate the above equation ; we get

dfi I I I

77= db 2n ^1 sin* - iSo — sin*- iS,
dt X 2 • 2

where fio is the maximum amplitude of the vibration.
Putting

sin }4fi = sin J^o sin <p,

and changing the variable from fi to <p, we get

nt^ r . "^^ = F(<p, Jfe), (2)

Jo v^ I — Jk' sin* <p

n^ = F(^.*) =K(k)

where k = sinj^o and F(ip, k) is the elliptic int^^al of the first kind.
Hence, if 7* be the period of oscillation,

4
or

J, _ 4Kik)
n

Now

cos 3^ = dn-nt;

.*. cos iS = 2 cos* }4P — I = 2dnhit — i.

Hence if Im and / be the intensity of magnetization as affected by
the thermal motion and that at absolute zero respectively, we have

'-U

I cos fidt

VOL.X.1

Na6. J

FERROMAGNETIC SUBSTANCES,

731

4/ r^* In r^*^'*

[2E{flmK,k) - K{k)],

Kik)

where E is the elliptic integral of the second kind.

Hence

I^ _ 2E(amK, k) _ _ 2E{k) _

(3)

The calculation of the ratio presents no difficulty. In the following
table and Fig. 22, the values of Im/I for different values of fi^ are given:

^0.

fmjf.

fio>

«.

/ml/.

0**

0.000

1.000

100**

0.766

0.352

20**

0.174

0.970

120**

0.866

0.126

40**

0.342

0 882

140**

0.940

- 0.108

60**

0.500

0.742

160**

0.985

-0.340

80**

0.643

0 562

180**

1.000

-1.000

Next, suppose iS > t; then the vibration changes into the revolution,
but its angular velocity is not uniform. As before, we have

= — n* sin jS

If for /3 = o,

I idfi\^

"XU)^ 2nV, where t;* > i,

Fig. 22.

then

putting P — 2<p and i/v* = k, we have

dtp

I — Jfe* sin' <p

= - dt'
k '

k Jo v^i _ k* sin*

(4)

732 KOtARO HONDA AND JVNZA dKUBO. [

For / = o and T, let ^ = o and x respectively, we have

dip

n r (

k^ J. x/r^

k^ sin* ip

Now

d<p

v/l -

- k* sin* <f

•
• •

T =

= V

.'. COS /S = 2 COS* §/3 — I = 2Cn^r * — i :

If we put X = (n/*)/, dx = {nlk)dt.
If / = o, then jc = o; if / = {2k/n)K{k), then x = 2iS:(i)

= ^^ { ^,[£(a«-2ii:(*)-*) - k'*-2Km - 2i(:(*) }

_ E{am-2K{k)-k) _ / *;^ \
k*K{k) Vk* ■'"V'

where ifc' is the modulus complementary to k; but

am'2i(r(Jk) = t;

Since k = i/v^ and v may take any value from i to oo , jk* can vary from
I to o. It is evident that so long as the angular velocity of the molecules
at /3 = o is not infinitely large, this velocity is not uniform, so that Im
does not theoretically vanish unless t;* = oo . This result is also evident
from the above relation.

In the following table, the values of /«// corresponding to the different
values of k are given:

VOL.X.1

Na6. J

FERROMAGNETIC SUBSTANCES.

733

tA.

UiL

<e.

t««.

/«//.

1.000

-1.000

0.643

2.410

-0.090

0.985

1.030

- 0.385

0.500

4.000

- 0.030

0.940

1.130

- 0.252

0.342

8.550

- 0.027

0.866

1.1335

-0.168

0.174

33.450

- 0.020

0.776

1.690

- 0.122

0000

-0.000

The relation between /«// and v* is also shown in the following figure.
As 0 or v* increases from o, the magnetization diminishes at first slowly
and then somewhat rapidly; in passing through /3o =?= 131^1 it vanishes
and changes its sign. With a further increase of /3o or t;*, the magnetiza-
tion increases negatively and at /3o = ^1 W^ becomes — i. Afterwards,
the magnetization rapidly decreases in absolute value, tending asymp-
totically to the value zero, as rf' approaches to « .

Now we find experimentally no evidence that the magnetization be-
comes negative at high temperatures, though the field is very strong.
What is then the cause of the discrep-
ancy between the theory and the experi-
ments? The cause is obviously to be
sought for the fact that in the above
theory, we have assumed no transfor-
mation either in the molecules or in their
mutual configuration. It is certainly
true that the above effect plays a part pig. 23

in changing the magnetization at high

temperatures. Probably in a value of the amplitude jSo, which is far
less han 131**, a gradual A% transformation will begin to proceed in the
substance, and consequently the substance is changed into the para-
magnetic state as conceived by P. Langevin.^

A few years ago, one of the present writers published a theory of
magnetism,* which is based on the Langevin theory of paramagnetic
gases; the theory connects the ferromagnetic and paramagnetic sub-
stances and coincides with the Ewing theory for the former substance.
It may be summarized in the following words: The form of the molecules
of a ferromagnetic sybstance is nearly spherical and consequently the
effect of thermal impacts in rotating the molecules is very small in
comparison with the mutual action; while in the case of paramagnetic
substance, the molecules have an elongated or flattened form, so that
here the effect of mutual action is very small compared with the rotating

» p. Langevin. Ann. de chem. et phys. (8), 5, (1905), 70.
* K. Honda. Sci. Rep., 3 (1914). 171.

734 k6TAR0 HONDA AND JUNZ6 Okubo. [^S

effect of thermal impact. The transformation of a ferromagnetic sub-
stance to a paramagnetic at high temperatures is by this theory explained
as a consequence of the gradual deformation of the spherical molecules
with the rise of temperature. The heat evolved or absorbed during this
transformation is considered to be the energy of transformation and that
imparted to the molecules to cause their rotational vibrations.

The above theory accords with the result of the present investigation.
On the other hand, the simple theory of the revolution of molecules is
not solely sufficient to account for the disappearance of magnetism at
high temperatures.

Next we shall consider the effect of temperature on magnetization in
the light of our theory of molecular magnetism. In weak fields, the
temperature affects the magnetization in two opposite ways; that is,
the first effect, which exists in all fields, is to diminish the magnetization
on account of the rotational vibrations of the molecules, and the second,
which is noticeable only in weak fields, is to increase the magnetization
by virtue of the abrupt turning of the molecules towards the field due to
heat motion. The observed change of magnetization at high tempera-
tures is the sum of these two effects. We shall firstly consider the first
effect from the standpoint of our theory of magnetism.

If the thermal agitation be zero, molecular magnets in each complex
will take a common direction determined by the external and internal
fields. Suppose this direction to make an angle B^ with the field. In
virtue of the thermal energy, they will in an actual case execute transla-
tional and rotational vibrations about their mean positions. The ampli-
tude of their rotational vibrations will actually differ from one magnet to
another; but as the first approximation, we may consider their mean
value to be jSo. Since, in each complex, the molecules exert their mutual
action on each other, the rotational vibration of molecules with the same
phase takes place more easily than in the case of those with arbitrary
phases. Hence in a stationary state, we may, as the first approximation,
suppose that all the magnets in each elementary complex oscillate with a
common phase, but that the phase of the oscillation differs from one
complex to another.

Consider at first the case, where the external field is very small as
compared with the internal; neglecting the couple due to the former
field, the equation of motion becomes

K-^^ -2Ar sin 4(^0 + fi). (6)

As ^0 is very small in weak fields, we may neglect it in comparison with

XS"^] FERROMAGNETIC SUBSTANCES. 735

fi. Hence, putting n' = {2Ar/K), we get

(Pfi

5? = -^'sin4^.

Now, let sin 2/3 » sin 2/3a*sin ^ =3 ife sin ^, and change the variable from
fi to ^, we get, after integration,

Jo v^i — Jfe* sin* ^

Let for / = o, iS = o .*. ^ = o; for / = 7/4, fi - fio .'. ^ = ^2.
Hence

„f = f(f.*)-i5:(fc)

^ _ 4K(k)

Now

sin <p = sn-nt,

sin 2iS = sin 2/3o sin ^ = k-dn nt\
.". cos 2/3 = dn-nt.

Hence the mean effect of a molecule making initially an angle a with the
field in the direction of the latter will be given by

I C
Mm^j^j Af COS (a — fi)dt

= ^.,. I COS a I COS j8d/ + sin a I sin fidt \ ,

where Mm and M are the magnetic moment of a molecule as affected by
the thermal motion and that at absolute zero respectively. But,

ll + COS2iS li

cos^ = ^ 2 = W"

. |i - cos 2/3 ll

sin^^=.^ =^-

+ dn nt

Since dn nt is an even function and its period 2K, we have, putting

X ^ nt

M cos a C^

Mm = , \ ^i + dnx dx

"[x)Vo

Mm ^2 KiGMi + *0)) , .

• (7;

JW cos a \/i ^ k' ^W

736

k6tarQ bonda and juNzd 6kvbo.

M cos a is the magnetic moment in the direction of the field. Thus the
ratio MmfM cos a for each molecule is a constant depending on /So-
Hence if /« and I represent the intensities of magnetization with and
without the thermal motions respectively, we have

Im_^i+k' K{kKi + k'))
I

K{k)

(8)

The ratio gradually decreases with the increase of /3o or of Jfe; for Po = t/4,
it becomes i/v^2 = 0.7071 • • • . U fio increases beyond t/4, the vibration

changes into a revolution and the mean eflFect of
magnetization vanishes; because in the present
case, the external field is neglected and the mo-
tion governed by the internal resisting force A
sin 46 with a period of T/4.

In the following table and in Fig. 24, the values
^^ of the ratio for different values of fio are given to
show how the magnetization diminishes with in-
creasing Po.

•A-

10* to* SO" io* so*
Fig. 24.

^0.

UII.

Ull.

U\l.

^0.

/«//.

5''

10**

1.000
0.992
0.985

15**
20**
25**

0.966
0.955
0.938

30**
35**
40**

0.920
0.895
0.861

45*

0.707

Thus the ratio gradually diminishes with increasing /3o up to /3o = W4t
where it suddenly vanishes. As we have already remarked, the diminu-
tion of magnetization with the increase of /3 would also be accelerated by
the A\ transformation, so that the fall of the curve with increasing ^
must actually take place at a smaller value of jS than T/4.

We shall next consider the second eflFect of temperature, which in-
creases the magnetization in weak fields. If the thermal motion be
absent, that is, at the absolute zero, the orientation of the equilibrium
of a complex, whose magnetic axis making initially an angle a with the
direction of the field, is given by

h sin (a — ^0) = sin 4^0;

hence if h be given, the relation between a and ^0 can easily be found by
the graphical method. If for a complex (a), ^0 + /3o > W4» then the
complex will oscillate about its mean orientation ^0; on the other hand,
if ^0 + iSo > W4, the. complex will undergo an abrupt turning and take a
position, as if the initial orientation were a — (t/2), causing thereby an
increase of magnetization. Hence, even in weak fields, where at absolute

XS^ft^l FERROMAGNETIC SUBSTANCES. 737

zero, there is no complex which abruptly turns in the direction of the
field, the complexes will more and more begin to make an abrupt turning
with the rise of temperature.

If there is no thermal motion, the reduced intensity of magnetization

is given by

I f

i = - I cos (a — 6o)da,

where a and ^o are connected with each other by the foregoing relation.
This relation for h = 0.5 is shown
graphically in Fig. 25. If iSo be given,
we can find from the above figure
the limits or the range of a, for which
the complexes make an abrupt turn-
ing toward the direction of field.
Let ai and at be such limits, then i is given by

= - j I cos (a — e)da + I cos (a — e)da +1 cos (a — e)da \ ,

It may also occur that some complexes, whose direction of magnetic
axis lies between a\ and aj, make the abrupt rotations twice or thrice;
in such cases, we must take for the limits ai — x and aj — x or ai — (3/2) x
and at — (3X/2), etc. In this way, under a given field, the value of i
corresponding to different values of /3o can be calculated. If we multiply
these values of i by the ratio

I x/r+k'' K{k) '

which represents the mean effect of rotatory vibrations, the resultant
intensity of magnetization will be those as affected by temperature.

Fig. 26 shows the result of our calculation for
^1 A =0.5; the ordinate represents the magnetization

••«| (\ in question and the abscissa the angle fio' The

temperature is obviously some function of Po in-
creasing with it. If we consider fi = x/4 to cor-
^o respond to the critical point, the course of the

* **° ^\^ ' *^ curve is quite similar to that obtained by J. Hop-

F 26. ig. , ^ ^ «-

kinson for a very weak field.

If h gets greater, the increased number of complexes turns abruptly

towards the field, even if there is no thermal motion; and consequently

the increase of magnetization due to the thermal vibration becomes

always less. In a sufficiently strong field, where all the complexes

0.4

738 k6TAR0 HONDA AND JVNZA dKVBO.

have finished their possible abrupt turning, the effect of temperature
in increasing magnetization must vanish, and there exists only the effect
of diminishing magnetization due to rotational vibrations. Thus the
effect of temperature on magnetization is explained by our theory, at
least qualitatively.

In the above calculation, the At transformation was not taken into
account. This transformation obviously affects in reducing the magne-
tization at high temperatures.

The theory so far explained strongly confirms the general view that
the magnetic phenomena are really due to the rotation of the molecules
about their own centers. This fact has an important bearing to the molec-
ular structure of ferromagnetic crystals, the discussion of which will be
given in a next paper to be published shortly.

§ 8. Molecular Field Introduced by Prof. P. Weiss.

Lastly the molecular field introduced by Prof. P. Weiss* will be con-
sidered in the light of the present investigations. According to him,
it is a uniform field acting on each molecule of a ferromagnetic substance,
its magnitude being assumed to be proportional to the intensity of
magnetization and having an enormous value amounting to several ten
millions of gauss. This molecular field was introduced by Weiss to
extend Langevin's theory of paramagnetism to the ferromagnetic sub-
stances; one of the present writers* has however shown that the same
extension can be made quite naturally by considering the molecules of
the ferromagnetic substances to be nearly spherical in form. The intro-
duction of the molecular field into the theory of magnetism meets with
great difficulties; namely his theory cannot explain very fundamental
and important facts in the theory of magnetism, such as the curve of
magnetization and hysteresis phenomenon.

The evidence, which Weiss sets forth as proof of his theory, b:

(i) The explanation of the magnetic properties of magnetite and
pyrrhotine by means of the demagnetizing field.

(ii) The existence of the corresponding magnetic states in ferromag-
netic substances.

(iii) The applicability of the relation

x{T — $) = const,

where x is the specific susceptibility at a temperature T higher than the
critical temperature $.

1 Conf^rrence i la Soc. francai. de Phys.. April 4 (1907). Arch, des Sci., No. 5, 31 (1911),
401.

* K. Honda. Sci. Rep., 3 (1914). 171.

JJSJ*^] FERROMAGNETIC SUBSTANCES, 739

(iv) The change of specific heat in the critical range of iron, nickel
and magnetite.

In explaining the magnetic properties of crystals, P. Weiss was led to
assume a uniform demagnetizing field of considerable magnitude. In
addition, with some improbable assumptions, he explained the com-
plicated magnetic properties of crystals; but we have shown in a paper, ^
that these properties can be very simply explained without assuming
any demagnetizing field.* Hence (i) can not be considered to support
his theory.

Secondly he obtained from his theory the relations:

>, = - 7- and -7- = coth a ,

u a lo io A

where T and 6 have the same meaning as before and a is a quantity
depending on the nature of the substance. If we eliminate a from these
equations, we obtain a relation giving a dependence of //Jo on I/$; this
relation is independent of the nature of the substance and therefore
called the relations for the corresponding states. This consequence affords
a means of verifying his theory. Weiss showed that this relation holds
good in high temperatures for magnetite and ferronickel, but in low
temperature the deviation between the theory and the result of the ex-
periment is considerably great. He also remarked that for iron and
nickel the agreement is only qualitative.

We have also examined the above relation for iron, nickel and nickel
steels of different compositions. For this purpose, it is necessary to
find the saturation value of magnetization at the absolute zero from the
observed values at low temperatures. We have here two methods:
Firstly, if we assume the above relations to hold good at least from the
observed lowest temperature to the absolute zero, we can find the value
of /o from the known values of J, T and 6, Secondly, we may also find
Jo by extrapolation from J, T curve actually observed. These two
methods do not give the same result. We found therefore two values
of Jo and calculated two sets of values of J/Jo and TjB for each specimen.
In our calculation, we availed ourselves of the results of experiments
made by Mr. S. Shimizu and one of the present writers' for Swedish
iron, nickel and nickel steels of 30, 36, 48, 50 and 60 per cent, of nickel.
The experiment was made at different temperatures ranging from liquid
air temperature to those above their critical points, and under constant
fields up to 700 gauss. For these specimens, the magnetization at liquid

»Sd. Rep., S (1916), 153-

* Jour. Coll. Sci., 20, Art. 6 (1904).

740

KOtarQ HONDA AND JUNZO OKUBO.

air temperature nearly attained its saturation value in the highest field
just referred to. The results are graphically shown in Figs. 27, 28, 29, 30.
The broken curve in each figure represents the theoretical one, while
other curves are the observed results. From these figures, we conclude
that the relation for the corresponding state is here only qualitatively
satisfied. Hence we can not regard the above relation as a confirmation
of Weiss*s theory.

Thirdly, Weiss obtained from his theory a relation

x{T - ff) ^ const.

One^ of the present writers made however a thorough investigation of
this subject, and showed that the relation is approximately true for iron.

H«700 ^^

Fig. 27.

Fig. 28.

nickel and cobalt and fails to be applicable in the case of magnetite.
He also showed that this relation can be obtained as a special case from
his theory, which does not take any account of the molecular field.

Fig. 29.

Fig. 30.

Hence as evidence for the existence of the molecular field, the above
relation has a little importance.

Lastly the change of specific heat at critical range of ferromagnetic
substances will be considered. It was shown by P. Weiss and P. N.

1 K. Honda, Sci. Rep.. 3. 1- c.; Sci. Rep.. 4 (1915). 248.

Vol. X.1
No. 6. J

FERROMAGNETIC SUBSTANCES,

741

Beck^ that the specific heat of iron, nickel and magnetite considerably
increases in the critical range. As however these metals evolve heat
by cooling through the critical range, what they measured is not properly
termed the change of specific heat by temperature, but the quantity of
heat evolved during the transformation' as measured calorimetrically.
This heat evolution was early measured by Pionchon,' Standfield,* and
recently by Meuten.* Weiss explained the heat evolved or absorbed
during the transformation as due to magnetic energy. Thus he calcu-
lated on one hand the change of magnetic energy per degree at different
high temperatures, using Curie's result on the magnetic measurement
at high temperatures, and on the other hand, in codperation with P. N.
Beck, he measured calorimetrically the heat evolution at high tempera-
tures up to the critical point. In this way, the change of magnetic
energy dcm per degree and that of the specific heat dc were compared
with each other for iron, nickel and magnetite; the results of his calcula-
tion are given in the following table:

Subfltance.

N.*

ac.

iC^,

753*'C. •
376**

588**

3,840
12,700
33,200

1,700
500
430

0.112
0.027
0.050

0.136

0.025

Fe,04

0.048

The agreement between 6c and dc^ is apparently as good as we can
desire. But it should be remarked that the thermomagnetic properties
of the ferromagnetic substances, and therefore the values of N, vary for
different specimens of the same metal, as the following table shows:

Substance.

A^ (Curie).

A'(Hond«,Tak«fi).

3,840
12.700
33,200

5,910^

10,730*

Fe,04

37,200-10,600

For magnetite, the quantity x{T — $) is far from being constant,* so
that N varies considerably with temperature. If we use the values of

» Jour, de Physique. 7 (1908), 249.

* K. Honda. Sci. Rep.. 4 (191 5)* 169.

* Ann. Chim. Phys.. 6th series, II. (1887), 33.
*Femim, i (1912). i.

» Jour. Iron and Steel Inst.. No. 2 (1899), 169.

* iV^ ■■ coefficient of molecular field. The change of magnetic energy per degree ■*

i ^(NP) "NI^.
2 di^ di

^ Sci. Rep.. 4 (1915). 261.

* Sci. Rep.. I (1912). 229.

* Curie, Oeuvres (1908), 322; H. Takagi. Sci. Rep.. 2 (1913). 117; P. Weiss and G. Foex,
Arch, des Sci.. 31 (1911). 89.

742 KOTARO HONDA AND JUNZA OKUBO. [^S

N given in the above table for the calculation of hcm, the deviation
between the theory and the experiment becomes considerable. In the
calculation of Weiss, the data for magnetic and calorimetric measure*
ments belong to different specimens, and therefore the coincidence in his
case may be accidental. It is, however, a remarkable fact that the two
quantities hcm and be, which are obtained from the quantities of quite a
different nature, coincide with each other at least in the order of magni-
tude, and therefore this instance may be r^arded as the most favorable
case put forward by Weiss. But the heat evolution or absorption in
the critical range can also be explained by another theory, as was actually
done by one of the present writers, and again, as shown in the present
theory, the principal features of magnetic phenomena, that is, the
magnetization curve, the hysteresis phenomena, and the temperature
effect on magnetization are satisfactorily explained without assuming
Weiss's molecular field ; hence the necessity for assuming the molecular
field will not only disappear, but the difficulties involved in assuming it
remain undiminished as before. Hence, it seems to us that the existence
of the molecular field put forward by Weiss b not consistent with the
observed facts.

Sbndai, Japah

1Q17.

jJS"6^'] HEAT CONVECTION IN AIR. 743

HEAT CONVECTION IN AIR, AND NEWTON'S LAW OF

COOLING.

By Walter P. White.

THE investigation here presented deals with convection in narrow
layers of air, and was originally undertaken in order to get data
to use in designing calorimeters. After considerable work had beeh
done it was learned that a more comprehensive investigation^ of air
convection was already in progress elsewhere, and it seemed more fitting
to avoid anything like an encroachment upon this other work. Our
investigation was therefore made less complete than might otherwise
have been the case, although it was extended to some points for whose
study our type of apparatus seemed to be especially well adapted.

The special bearing of convection on calorimetry has to do with the
**law of cooling" of the calorimeter. It is convenient in practice,
though by no means necessary,* that this law should be '* Newton's
Law," that is, that the thermal leakage should be proportional to the
thermal head} (temperature difference) which causes it. In so far as
the leakage is due to conduction or radiation, it will, as can be readily
shown from familiar laws, conform substantially to Newton's Law.
But convection is due to air currents whose temperature and velocity
are both affected by the thermal head, and which therefore tend to
convey heat at a rate often more nearly proportional to the square of that
head. To convection is due most of the observed variation from New-
ton's Law. A knowledge of the magnitude and character of the convec-
tion effect, therefore, was expected to facilitate the designing of more
satisfactory calorimeters. The application of the present results to
calorimetric practice will be made in another paper in the Journal of
the American Chemical Society. A preliminary statement of results and
application has already appeared in this journal.*

Attempts to study convection have of course been made, usually, it

» Since presented in part, " The Testing of Thermal Insulators," H. C. Dickinson and M. S.
Van Dusen, A. S. R. E. Journal, 3-5, 1916.

« See, e. g., "Some Calorimetric Methods," Walter P. White, Phys. Rev., 31, 553-557. ipio.

•There are different kinds of "temperature dififerences" entering into thermal problems,
so that it seems desirable to try to distinguish them. The term thermal head has been
selected for simultaneous temi>erature difference, which causes heat flow.

* Proc. Am. Phys. Soc. This journal. 7, 682. 1916.

744 WALTER P. WHITE, [^SS

would seem, with reference to bodies of different kinds. In the present
case it seemed that since the convection was a matter of the air, simplicity
and definiteness in the air spaces, rather than in the solid bodies, should
be sought, and the problem was decomposed into that of a thin vertical,
and that of a thin horizontal air layer.^

Vertical Convection.

The vertical convection received the most attention. Direct observa-
tion (with the aid of smoke) of the air currents, as well as their thermal
phenomena, show that, for spaces usual in calorimetry, the flow, up one
side and down the other, is approximately "stream line flow," that is,
free from eddies.' Assuming it to be entirely so, we obtain the following
deductions, for surfaces whose height is not too small.

1. The currents, while moving parallel to the surfaces, will, in general,
carry no heat either to or from them. The ascending or descending
currents, on first striking the surfaces, will take up or give out heat near
the edge which they first strike, but will, as they move up or down, soon
acquire nearly the same horizontal temperature distribution as might
prevail in still air. Hence the total effect of convection will be inde-
pendent of the height, that is, the convection effect per unit area will
vary inversely as the height.

2. Since stream line velocity is proportional to pressure difference,
the speed of the currents where it prevails will vary as the difference
of density in the air, and therefore as the temperature difference between
the two surfaces.

3. The velocity of the currents, and therefore the convection effect,
will vary as the third power of the distance between the surfaces. This
results from the same reasoning which gives the familiar rule that in small
tubes the velocity varies as the fourth power of the diameter. It is true
not only for uniform pressure, the case usually considered, but also for
a pressure difference varying with the density of each vertical layer of
air, that is, varying regularly across the space from one surface to the
other.

4. The convection will diminish with the mean temperature, on account
of the increase in viscosity and rarity of the air.

^ If the layers are not thin, and surround the same body, they will doubtless affect each
other. This investigation was based on the notions : (i ) That this mutual effect was negligible
in the case practically presented by the calorimeter, and (2) that any investigation should
begin with the simpler cases.

* By early calling my attention to these facts. Dr. H. C. Dickinson and Dr. E. Buckingham,
of the Bureau of Standards, undoubtedly shortened the time required by the present
investigation.

•L.X.1

>. 6. J

Vol. X.1
No.

HEAT CONVECTION IN AIR.

745

5. The convection currents themselves tend to destroy the tempera-
ture difference which causes them. The faster the air moves, the farther
up the plate will it go before reaching equilibrium temperature, the
lower will be the mean temperature of the heated layer whose difference
of density produces the flow. Hence for actual finite surfaces the currents
will not increase with distance, etc., as fast as paragraphs i, 2, 3, would
indicate.

The results, to the limit of their precision, proved to be in harmony

with these deductions, except that they also revealed another effect

which in some cases modifies perceptibly the amount of convection, and

masked effect number 5, though without indicating that deduction 5 is

not true.

Apparatus and Methods.

The very simple air spaces called for by the plan of work were provided
by using for the solid bodies plates of metal, each of which presented,
practically, but one surface to the air. j^

The heat was received by a flat, silvered
plate of copper; it flowed from the wall
of a rectangular copper box inclosing ^
each plate, which, tight soldered, was
immersed in an electrically heated bath
of water or kerosene (Fig. i). The
change in plate temperature measured
the heat flow; the thermal head was the
temperature difference of box wall and
plate. These two quantities were con-
veniently measured by thermocouples
soldered to the plates; a single differen-
tial measurement gave the thermal head.
Time was the only other quantity to be
measured.

Usually the temperature difference was

kept constant. The plate was thus

Fig 1
warmed regularly, and when the faster

, . r 1 • Convection apparatus. P, receiving

rates were used a series of determina- ^^,^^^ s. sUk suspension, ir. wooden
tions covering 30** in mean temperature locating pins. r. thermoelectric tem-

was obtained in a few minutes. perature measuring wires.

In a preliminary series the receiving plate was either a horizontal
disc or a small hollow cylinder, inclosed in another cylinder 5 cm. in
diameter. This series gave an apparently perfect confirmation of deduc-
tion I, above, but was not as accurate as the later work, and will not be

746

WALTER P. WHITE.

mentioned again. Subsequently flat plates were used, 2 mm. thick and
8 cm. square, or about half the height of an average calorimeter. These
were in pairs, back to back; the thermo-junction was between the two,
and was thus protected from any direct effect due to the air. The junc-
tion on the can wall was sometimes outside, in a kerosene bath, and at
other times inside. This change of position appears to eliminate the
slight systematic error arising from the difference between the tempera-
ture of the plate itself and of its junction with a wire running out into
fluid of slightly different temperature.

Three different thicknesses of air gap were tried with the square plates :
8 mm., 12 mm. and 24 mm. No guard rings were used, and the edges
of the central plates did not reach the walls of the case.

The work at the smaller thermal heads tested the precision of the
apparatus to the full. Some especially good constantan wire, 0.125 mm.
in diameter (Number 36), was available for the thermo-couples. The
most important wires were tested and selected, and showed in electro-
motive force against copper no variation over 0.0002. Readings were
made to o.i microvolt, which corresponds to 0.0025* with a copper-
constantan couple, and comparisons between heat transfer values, con-
taining the errors of 4 observations, very rarely differed by over 0.2

Fig. 2.

Arrangement of apparatus, tank, and stirrer one-fifth actual size [or, scale, one-fifth size].

H, annular electric heater.

microvolt for slow, and 0.4 for the more rapid, temperature changes,
equivalent to i per cent, and 0.2 per cent., respectively.

Since it is the change in plate temperature which is used in the calcula-
tions any constant error in this reading is immaterial, while any variable
error would be part of the visible accidental error. A very small con-
stant error in the thermal head, or temperature differences of the plate,*
however, might have caused a systematic error in the results. Hence
such error was repeatedly looked for by making determinations with
the head nearly equal to zero. The temperature change of the plates

1 Unless it should be a proportional error, constant for all the observations. This would
be unimportant, since it would still leave the results consistent with each other.

^^] HEAT CONVECTION IN AIR. 747

for zero difference was then almost invariably less than 0.0025", the
observational error, in periods of from 10 to 30 minutes. Control
observations of temperature were made at various points on the outside
can, and these indicated that temperature inequalities here were quite
negligible. This uniformity was secured by the powerful stirrer, which
was 12.5 cm. in diameter, and occupied about a third of the oblong tank
(Fig. 2), Com[>ari5ons of the temperatures at
different parts of the central or receiving plates
showed, first, that the difference of temperature
between top and bottom of the plate which is to
be ex[>ected from the action of the convection
current was present, and was, for the 12 mm.
gap, 0,005 of the difference between plate and
can; and second, that this temperature distribu-
tion was nearly established in 15 seconds. There
is therefore no doubt that in the results as ob-
tained there was present no error due to a fail-
ure to establish equilibrium soon enough.

The plates were supported by silk thread, and
kept central by wooden pins, 3 for each pair,
screwed through holes in the plates. The pins
were 2 mm. in diameter, and their heat conduc-
tivity was estimated to be 0.008 that of the air.
It seemed improbable that they caused any dis-
tortion in the relative values of conduction and
convection. But since a very unexpected rela-
tion appeared among the results, so that it
seemed best to avoid even remotely possible
chances of error, one pair of plates (i3 mm. gap)
was given an all-silk suspension, as shown in Fig. _. ,

3. At the same time great precautions were ^^^ ^^^ ,^.^„^„
taken against distillation of vapor. (The dis- ippaiatus. SUk corda run
tillation of only 0.2 mg. of water would have ■l'™™ the side tubes and can

raised the plates 0.005°, and this rise, if the dis- ^ '^""™ "*"" " ""' "°"
, , after the case ia soldered to-

tillation took ten mmutes, would have had a gether. c. glass tubes ce-
signiffcant effect on the results.) Hence the mented on for running dry
bath fluid was changed from kerosene to water, ^ f^rough the case. K.

, „ , 1 , . lumps o[ <%menC.

m order to use a fluid which could be thor-
oughly removed from the inside of the can, and whose presence there
could be detected with certainty. The can was dried for from 3 to 9
hours by a current of air after each day's work, and the water removed

75°

WALTER P. WHITE.

OXMS

^20*Hood

(X040

lO'Head

•
535

^Vv^S'H

ead

I
0030

a025

9, "

o
o o

c
c

500 1000 1500 2000

Mean Temperature ,
Microvolts

Fig. 6.

Results for 24 mm. gap. Half circles are val-
ues for a head of for 0.5^.

series with 12 mm. air gap repre-
sent different arrangements of the
apparatus.

Figs* 4» 5f 6 serve as a presenta-
tion of the original results, give
an idea of the agreement ob-
tained, and show the variation of
heat transfer with mean tempera-
ture. They are a little mislead-
ing as to the variation with ther-
mal head (difference of tempera-
ture), since they are derived di-
rectly from results expressed in
microvolts, and the microvolt
reading of a copper-constantan
thermoelement is not exactly
proportional to the temperature
in degrees. This anomaly is re-
moved in Table I., and in Fig. 7,
derived from it, which show the
value of k (total heat transfer
divided by head) in the usual
C.G.S. units for a mean temper-
ature of 900 microvolts (about
22.5**) for each of the three air-
gap widths. The difference be-
tween I® and 2® for 12 mm. is just
about the experimental uncer-
tainty, or 5 per mille, as appears
from Fig. 5. The exact position
of the line for no convection is
therefore a little in doubt, for this
and the 8 mm. gap also.

The results for these two gaps
are in agreement with the original
deduction (3) that the convection
varies as the cube of the gap
width, indeed, the ratio for differ-
ent thermal heads evidently varies
by less than the very small differ-

Na6. J

HEAT CONVECTION IN AIR.

751

ence between the values for the 12 mm. convection. This extreme con-
cordance may of course be partly accidental, and the range of width is
too small for the results to be considered as a demonstration of any law,
but the agreement is clearly all that could possibly be expected.

Table I.

Heat Transfer, in the Usual C.CS. Unit, i. e.. Calories per Second per Degree of Thermal Head

per Square Cm. of Flat Surface, at 22.8° Mean Temperature,

Where two values are given they show the range among determinations with different
methods of getting the temperature of the outer plate. It will be seen that the value of the
convection is practically unaffected by this difference of method.

Tit ^Plll • 1

8 Mm. Gap.

IS Mm. Gap.

94 Mm. Gap.

HMd.

Total.

Convec-
tion.

ToUl.

Convection.

Total.

Convection.

.99**

1.98**

4.95**

9.89*'
19.76*'

r.OOO 109
I 110

.000 111

r.OOO 112

\ 113
.000 116

.000 001

.000 003
003

.000 007

.000 083 9\
.000 084 8/

.000 084 0\
.000 085 2/

r.OOO 086 6
\ 88 1

.000 093 7

95 2

r.OOO 107 7

I 109 4

.000 000 1
000 4

.000 002 8 1
003 7/

.000 010 \

.000 011/

.000 024 \

026/

.000 065

.000 090

.000 106
.000 126

over .000 025

over .000 040
over .000 060

After a certain critical velocity is reached, streamline motion passes
(suddenly in long tubes) over into turbulent motion, whose velocity
then increases as the square root of the temperature difference (that is,
of the driving pressure). Most of the 24 mm. gap results show an
approximation to this condition, and have therefore been considered
out of the range to which Deductions 1-5 fully apply.

The 12 mm. gap results show an unmistakable tendency for the con-
vection to increase faster than the temperature interval. This result,
contrary to the inferred law for stream-line flow (Deduction 2), was
unexpected, and the special precautions to secure accuracy, described
above in the section on apparatus and methods, were taken mainly to
be sure that no experimental error was responsible for the increase. That
the effect comes from an abnormally low value of the convection for small
intervals is unlikely. At least, I have been unable to imagine any cause
for such an action. A satisfactory explanation is to suppose that the
heat transfer for larger intervals is greater than that proper to stream-
line flow. The main cause for this excessive transfer seems to be an

752

WALTER P. WHITE.

[

incipient turbulence at the ends of the stream, which carries across
more heat, even though the velocity of the stream may perhaps be made
less rapid by it. Indeed, the sharp distinction usually said to exist
between stream-line and turbulent flow is characteristic only of long

tubes. In short tubes turbulence
enters more gradually, hence the
results here observed are appar-
ently only what should have been
expected. Again, the apparently
excessive diminution, noticed
above, of convection with increase
of mean temperature points
strongly to turbulence, for we
have something decreasing very
rapidly with increase of viscosity,
which is precisely what turbulence
would be expected to do. Dickin-
son and Van Dusen, also, by
measurements of air temperature
across the gap at the middle of
plates 20 cm. high, found irregu-
larities which seem to indicate in-
cipient turbulence, and this was

P > 10 19 CO

Th«rmoi Heod In DegKe«9 perccptiblc for 3, 15* difference

Fig. 7. and a 15 mm. gap.^

„ ^ ^ r r r .. r ... The tempcrature difference

Heat transfer factor aa a function of ther^
mal head. Series a and 6 both indicated for (0.OO5 of the thermal head) along

12 mm. the inner plate for 12 mm. gap,

mentioned above, was surprisingly
close (to about 10 per cent.) to that calculated, with some approximation,
from the convection heat supply and the conductivity of the plate. (The
formula derived gave the difference as nearly independent of the emissiv-
ity of the plate.) The calculation was made on the supposition that the
convection heat was all delivered at the edge of the plate (t. «., that De-
duction I is correct), the result therefore furnishes an independent con-
firmation of the reasoning and of the early experiments, not here given,
on which that deduction is based. For the 24 mm. plate the observed
difference was nearer half of that calculated. This result is consistent
with the supposition that with the turbulent flow at that gap width
part of the convection heat was .delivered elsewhere than at the edge of
the plate, though most of it near the edge.

* Presented orally to the Washington Philosophical Society, 1916.

ooootw

/___^

9 IM

8»2l]2.^2E<

"/^

^y<
<^^

C M

<^^

.0 10

•

Jf^

r ao

rW^

C ""
0

OOOOOM

•

JJ?J"(^"] HEAT CONVECTION IN AIR, 753

Various more or less obvious extensions of these experiments would

render the conclusions more complete, but for reasons already stated in

part it has seemed best to limit the present work to its original purpose,

of assisting calorimeter designing, which seems, in the main, already

accomplished.

Conclusions.

On the whole, the conclusions anticipated at the beginning of this
paper are supported, except that the convection investigated apparently
proved not to be of the pure stream-line type to which those conclusions
strictly apply, so that with it the convection transfer increases more
rapidly with temperature than it would do with pure stream-line flow.

This turbulence renders the convection effect more uncertain than it
would otherwise be and, especially, more dependent on the conditions
at the edges of the surfaces. An attempt at a theoretical treatment
would apparently encounter decided complications. The clearance at
the edges in the present case was made equal to the gap width for the
8 mm. and 12 mm. intervals, as the closest practicable imitation of
calorimetric conditions. In applying the results to calorimetric problems
it seems safe to assume that where the air flow is practically of the
stream-line type (i. e., for 8 mm. gap) the magnitudes here given would
hold with practical exactness, but otherwise (e, g., for a 12 mm. gap)
uncertainty as to the amount of turbulence would render conclusions
uncertain by an unassignable amount, which, however, would not be
a material detriment in calorimeter designing. The way in which the
rim of the calorimeter was disposed might make considerable difference.
In general, the smaller the variation from Newton's Law, the more
certain will be its determination by means of the above results.

The absolute value of the heat transfer without convection in Table I.
is evidently too high for conduction. This is accounted for by the effect
of radiation, and also by the end (or edge) effects. It has already been
said that not the slightest attempt was made to get correct values for
conduction. Results for the convection alone are probably much less
influenced by these disturbing factors.

Of the results of others little is directly comparable with those given

here. Most of the experimenters appear to have worked with cooling

bodies as a whole, and not with simple air gaps. L. Lorenz^ did a little

.work, and I. Langmuir^ made a more complete investigation, with flat

surfaces, but each used practically an infinite air gap, so their results

*"Ueber das Leitungsvermdgen der Metalle fur Wftrme und ElektrizitUt," L. Lorenz^
Ann. d. Physik, 13. 586 (1881).

""Convection and Radiation of Heat." I. Langmuir, Trans. Am. Electrochem. Soc, 23^
299 (1913).

754 WALTER P. WHITE.

do not seem comparable with ours. Dickinson and Van Dusen's* results,
however, were obtained with a range of air gaps which included ours.
For 24 mm. gap their agreement seems not unsatisfactory with the
present data and also with Deduction i. For 15 degrees head they get
the mean K per sq. cm. due to convection K = 0.000 030 for a height of
18.6 cm., and about 0.000 050 for 9.3 cm. against 0.000 050 for 8 cm.
interpolated from Table I. For 12 mm. and the same head, they get
0.000 008 for 18.6 cm. height, which, since the mean K is inversely pro-
portional to height, corresponds to 0.000 018 6 for 8 cm., practically the
same as 0.000 018, the value according to Table I. For a 9.3 cm. height,
however, they get 0.000 027, which is much greater than 0.000 018. The
discrepancy here with our result is evidently no greater than with their own
result for a different height and the same gap; that is to say, for 12 nmi.
gap their results do not accord with Deduction i, although for most gaps
they do.* Moreover, this disaccord can not be explained by supposing
that Deduction i is generally erroneous, and that turbulent flow occurs
all over the plate, for this would evidently produce the opposite kind of
disagreement. The explanation will doubtless be found through the
continuation of Dickinson and Van Dusen's investigation, whose inter-
ruption is greatly to be regretted. The occurrence of such an apparent
anomaly in so skilful work is at any rate an indication of the variegated
possibilities of convection phenomena. Aside from it, the agreement
between that investigation and this is almost unexpectedly good, con-
sidering the precision of each and the radical differences which do in fact
exist between the methods used. And even that anomaly does not ap-
pear to affect materially the practical value of the present results, for
the following reason. One principal difference between Dickinson and
Van Dusen's methods and ours is that their central plates extended to
the edge of the air space, while ours did not. The turning point of the
air currents, and the turbulence which is probably greatest there, thus
come opposite a part of the central plates in their case, and not in ours.
Our arrangement, of course, corresponds more nearly to calorimetric
conditions. Moreover, the anomaly was observed for the rather large
head of 15**. That anomaly, therefore, occurred under, and was probably
dependent on, conditions not common in calorimetry. Furthermore,
fortunately for the present purpose, comparatively large discrepancies
in the convection constant correspond to small differences in gap width,
so that in practical application to calorimeters it is easy without appre-
ciable loss to make allowance for the possible uncertainty in the data.

* Loc. cit.

* This discrepancy was first demonstrated from the curves in the published paper, but
subsequently the authors very courteously placed more complete and accurate data at my
disposal, and these have been used in the above discussion.

^^^] heat convection in air. 755

Horizontal Surfaces.

The 8 mm. and the 12 mm. apparatus, with wood insulators, were used
horizontally. The convection was surprisingly like that for vertical
surfaces, in spite of the great difference in the character of the convec-
tion currents in the two cases. Of course convection operated on only
one of the two horizontal surfaces. The convection heat transfer was
a little greater than that from both vertical surfaces, but since convection
will only occur on one horizontal surface in a calorimeter also, the calo-
rimeter, for gaps not over 1.5 cm., can be handled well enough by taking
the convection per sq. cm. of e(ich horizontal surface as equivalent to that
from a vertical surface about 7 cm. high.

SUMBiARY.

In very narrow layers of air between vertical surfaces at different
temperatures the convection currents, in the main, flow up one side and
down the other, with eddyless (stream-line) motion. It follows that
these currents transport heat to or from the surfaces only when they
turn and flow horizontally, from which fact it follows, in turn, that the
convective heat transfer is independent of the height of the surface.
It is, according to the laws of eddyless flow, proportional to the square
of the temperature difference, and to the cube of the distance, between
the surfaces. As the flow becomes more rapid (e, g., for a 20® difference
and a distance of 1.2 cm.) turbulence enters, and the above relations
begin to change. The change is apparently gradual, and the present
results as well as some obtained by other experimenters are rather nega-
tive as to the possibility of expressing the flow simply for the correspond-
ing range of conditions, which covers those most usual in calorimetry.
The results, however, are sufficient to serve as a practical guide in
calorimeter designing. For the dimensions tested, convection in hori-
zontal layers was a little over twice that in vertical.

Geophysical Laboratory,

Carnegie Institution op Washington.
Washington, D. C,

Julys. 1917.

756 C. C. BI DWELL.

ELECTRICAL AND THERMAL PROPERTIES OF IRON OXIDE.

By C. C. BmwELL.

FURTHER data on electrical resistance and thermo-electric power
of specimens of iron oxide, both FesOs and FeiOi, together with
data on thermal conductivity of FejOj, have become available since the
publication of a report on the subject from this laboratory in July, 1916.^
Thermo-electric power and resistance data have now been carried to the
melting point of the oxides studied, approximately 1520° C, four
hundred degrees higher than was possible with the earlier apparatus.

Two methods of preparation were employed resulting in specimens of
quite different physical behavior. The first method and that employed
in the previous work consisted in first fusing a quantity of FesOa in an
arc furnace thus producing, as chemical analysis showed, a solid mass of
FeaOi. This was ground to an almost impalpable powder, compressed
by means of a hydraulic press, baked at a bright red heat for one hour or
more and then ground to the form desired, usually a rod some 15 mm.
long by 6 sq. mm. cross-section. The tips of the specimen were melted
by means of an oxy-hydrogen flame and platinum, platinum-rhodium
thermo- junctions wires ** frozen" in. The second method of preparation
consisted in working the molten oxide into the form and size desired and
fusing in the platinum junction wires as the specimen solidified.

Chemical analysis showed the specimens prepared by the first method
to have completely reverted, after the baking process, to FejOi. At
room temperature the resistance of specimens so prepared is beyond
10^ ohms. The specimens prepared by fusion without baking show a
resistance of but one or two ohms at room temperature The specimens
prepared by the first method are comparatively stable, the data being
approximately reproducible, run after run (see Fig. 2). (These specimens
could as well have been prepared by direct compression of the original
FejOa and baking without the fusing and resultant reduction. It was
not known at this time, however, that the subsequent baking caused the
reversion to FejOa.) The specimens prepared by the second method,
that is, the FeaO* specimens, are unstable, heating even to three or four
hundred degrees causing a large permanent increase in resistance on

» Phys. Rev., N. S., Vol. VIII.. No. i, p. la (1916).

nS"6^] iron oxide, ysy

cooling. Heating to 1400° C. causes a permanent radical change, the
specimen behaving thereafter in all respects like the baked specimens
(FeaOs). Chemical analysis of such a specimen showed an oxidation
to FejOs. The data for the fused unbaked specimens are therefore
very rough and not reproducible. The curves for these specimens
however serve to show the radical difference electrically between FcjOi
and FejOa (see Figs. 3 and 4).

Correction to a Previous Report.

As a result of the chemical analyses carried on in connection with
the present work it is desired to here indicate a correction to a previous
report, namely that on "Resistance and Thermo-Electric Relations in
Iron Oxide," published in the Physical Review, N. S., Vol. VIII.,
No. I, July, 1916. The specimens there reported upon are now known
to have been FejOs rather than, as stated, FeaO*. The specimens were
FeaOi at the start but the present work has shown that the baking in
air at a bright red heat always oxidizes the specimen completely to Fe^Oa.

Methods and Results with FEiOa.

The specimen, prepared by fusion and baking as explained above, was
placed in the end of a mall quartz tube of about one cm. inside diameter,
about the end of which were wound two turns of No. 20 platinum
wire (xx'). The quartz tube containing the specimen and the wires
(xx') was placed in the center of a platinum-wound tubular furnace
and resistance measurements taken by a fall of potential method as

Fig. 1.

indicated in the previous paper.^ To get thermo-electric power a tem-
perature gradient was established along the specimen by sending current
through the wires xx\ As soon as equilibrium was established the
thermo-electric power was observed. The heater circuit (xx') was*then
opened and, when temperature equality was reestablished along the
specimen, the resistance observations were repeated as a check on the
first readings. The readings are thus seen to be simultaneous to the
extent that it is possible to get simultaneous observations of two proper-
ties, one of which requires a uniform temperature, the other a temperature
gradient. From five to ten minutes were required for the establishment
of equilibrium conditions in the above cases.

* Loc. dt.

758 C. C. BIDWELL.

The observations, of course, give the thermo-electric power for the
oxide against platinum. This was found to be negative for the lower
ranges and, at 400° C, of the magnitude of 600 micro-volts per d^jee.
The sign was determined by means of the rule stated in a previous
paper,^ viz., if current flows from the oxide to platinum across the hot
junction, the oxide is said to be n^^ative to platinum, and the E.M.F. is
taken as negative.

Resistance measurements were plotted in accordance with the equation*

W is the resistance (specific resistance was not determined). T is the
temperature on the absolute scale; A and B are constants.

Observations in the past have shown considerable variation on succes-
sive runs in both resistance and thermo-electric power values. In order
to study these changes on successive heatings, a series of runs were taken
on the same specimen, the specimen not being mechanically disturbed
in any way during the whole series of observations. Four runs were
thus obtained, each run starting at a higher temperature than the pre-
ceding. Observations were taken with descending temperature steps.
The data are shown graphically (Fig. 2). Simultaneous resistance and
thermo-electric power runs are shown plotted together.

The transformation previously reported* at 710-730* C. is confirmed
by both resistance and thermo-electric power lines on every run. The
thermo-electric power line is straight up to the transformation point, but
beyond that temperature steadily and consistently deviates, showing two
reversals in sign. A second, very marked, reversible transformation is
revealed at 1320** C. by the behavior of both the resistance and thermo-
electric power lines. An inversion point for FcjOa at 1250-1350° C.
has been reported by Kohlmyer* on evidence obtained from cooling
curves. Kohlmyer also reports an inversion at 1028-1035** C. This
latter point was not corroborated by the present data.

The variations in the different runs are probably due to the dissociation
and recombination of oxygen on heating and cooling the specimen. The
amount of recombination on each occasion depends probably upon the
rapidity of cooling.

In the previous paper* a relation was thought to be indicated between

» Phys. Rev.. N. S.. Vol. III.. No. 3, p. 207.

■ For discussion of this equation see previous paper. Phys. Rbv., N. S., Vol. VIII., No. i,
p. 12.

» Loc. dt.

* E. J. Kohlmyer, Metallurgie. 6. 323-325 (1909).

* Loc. cit.

Fig. 2.

Resistance and thermo-electric power of a specimen of FesOi, — successive runs on the same
specimen. The resistance lines are plotted to the scales indicated to the right and along the
bottom. The thermo-electric power lines follow the scales indicated to the left ancf at the top.

^6o

C, C. BIDWELL,

(0 the dectron beat of dissociation (pfX>portioiial to the dope of tbe
resistance line) and the slope of the thermo-electric power line, viz.,
that an increase in (0 meant a decrease in the rate of change of thermo-
electric power and vice versa. The present data does not bear out this
idea but rather indicates no simple relation between these quantities.

Methods and Results with Fei04.

Specimens prepared by fusion without subsequent baking were found
upon analysis, as before stated, to be pure Fei04. These specimens
were ground to approximately the same nze as the baked specimens
and upon them the same electrical measurements were made. Since
oxidation goes on more and more rapidly as temperature rises, clear-cut,
definite results on temperature variation of the property under study
are not possible without control of the oxygen pressure. Certain general
information however can be obtained. The FesOi specimens are char-
acterized by low resistance (one or two ohms at room temperature),
with n^ative temperature coefficient, the resistance dropping to two or

Fig. 3.

Resistance and thermo-electric power of a specimen of Fes04 (showing oxidation at 1400° C.
to FesOt). Resistance is plotted directly against temperature.

three tenths of an ohm at 1200** C. The resistance curves (see Figs. 3
and 4) usually indicate a transformation between 600** and 800* C. On
cooling from 1400* C. the resistance increases rapidly and if the heating
has been sufficiently prolonged (one hour or more) the specimen then
behaves in all respects like FeiO«. The resistance on cooling to room

Vot-X.

Na6

IRON OXIDE.

761

temperature may be anything between the initial value of one or two
ohms and 10^ or more ohms depending upon the degree of oxidation.
Specimens which showed the complete change to the behavior of FejOj
specimens were found on chemical analysis to have the corresponding
composition, namely 70.0 per cent. iron. Approximately the same
thermo-electric power line was obtained with four different specimens.
The relation is indicated by two straight lines differing slightly in slope,
the transition from one to the other occurring between 700° and 800° C.
The values of thermo-electric power are small corresponding to the low
resistance of the specimen. After prolonged heating the line becomes
the typical thermo-electric power line for FejO$. Figures (3) and (4)

tjioti w wo' 835^ (000' laoe* Wrc

Fig. 4.

Resistance and thermo-electric power of a specimen of Fe904 (showing oxidation at 1400^ C.
to FesOt). Resistance is plotted directly against temperature.

show resistance and thermo-electric power changes on heating, for two
specimens of FesO*. The dotted lines indicate the behavior on cooling.
Recent observations of Sosman and Hostetter^ have shown the possi-
bility of solid solution in the system FejOr-FeaOi with all gradations of
ferrous iron from zero to 33.33 per cent. A more exact study of the
effect on the electrical resistance of the dissociation of oxygen, or the
variation of ferrous content, at a given temperature, is now being

attempted.

Thermal Conductivity of FEjOa.

Very little is known concerning the change of thermal conductivity
with temperature in the case of the so-called "variable" conductors of

1 Am. Chem. Jour., Vol. XXXVIII., No. 4, Apr., 1916.

762 C. C. BIDWELL. [^^

which iron oxide is a type. Even in the case of metals, data over wide
temperature ranges are quite meager. Therefore a study of the thermal
conductivity of iron oxide seemed to offer interesting possibilities of
information in this field. By the method described below it has been
found possible to obtain results through a temperature range of about
looo degrees. These results though admittedly rough are thought to be
significant. The absolute values are not of great impprtance since they
depend somewhat upon the density and composition of the particular
specimen and these factors are determined largely by the mode of
preparation and the heat treatment. The law of temperature change
however is significant sin<% it is probably not effected by the uncertainty
as to whether all the values are high or all low.

Powdered FeiOt, prepared by fusing pure FeiOi in a carbon arc, was
compressed in an iron cylinder The iron cylinder was 16.0 cm. long,
5.8 cm. outside diameter, 3.38 cm. inside diameter. At the center of
the cylinder was held an iron rod, one cm. in diameter, and wrapped in
several thicknesses of paper. This rod was allowed to pass through a
hole in the piston used with the hydraulic press and in this way the

Fig. 5.

oxide was compressed about it. After compression the whole arrange-
ment was heated to a bright red heat and the iron rod easily withdrawn
owing to the charring of the paper covering. This left a hole through
the center of the solidly packed oxide core extending the length of the
specimen. The central hole was designed to carry a heating coil (see
Fig. 5). This heating coil was of nichrome wire and was wound on the
inside of a quartz tube which was of such diameter as to fit the hole
closely. Junction wires of platinum and platinum-io per cent, rhodium
were placed at ^, S, Cand D (Fig. 5). The wires {ox A and B were run
through the central hole between the oxide and the quartz tube. The
fit was so tight that the junction enlargements were believed to be tightly
pressed against the inner surface of the oxide. Junctions C and D reached
the outer surface of the oxide through holes in the iron casing. These
holes were plugged with alundum cement. In order that a radial heat
flow from the central portion might be reasonably certain, the oxide and

uSrt^'] IRON OXIDE. 763

its iron casing were sawed through at M and N and then bolted together
again. A contact of this sort offers a high resistance to the flow of heat
across it. The length of this central section was 4.94 cm. The input of
energy into the section MN was obtained by measuring the current and
the potential drop through the heater, multiplying the pd. by 0.309, the
ratio of the length of the central section (MN) to the entire length (PQ).
The quartz tube containing the heating wires was packed with powdered
aluminum oxide to eliminate convection. The outer iron casing was
covered with about 5 mm. of alundun cement in which were embedded
the junction wires leading to D and C. This whole arrangement was
then slipped into the center of another iron cylinder approximately twice
as long as PQ and of such diameter as to fit the specimen closely. The
outside of this cylinder was also covered with a layer of alundum cement
and in this was embedded a winding of nichrome ribbon running the
whole length. Ordinary asbestos packing completed the construction.

The specimen could be heated uniformly to any desired temperature
up to 1050** C. Tests when the inner heater wires were carrying no
current gave uniform and steady temperatures after about five hours'
heating. The criterion was simply that all four junctions should read
alike. The final observations for thermal conductivity were made after
both the outside furnace and the inner heater current had been on for
some six to eight hours. Readings were begun after about five hours'
heating and then taken every half hour, the last two or three readings
usually being constant and indicating steady conditions.

The method above described is similar to that employed by Angell^
in measuring heat conductivity of nickel and aluminum
merely in that the shape of the specimen was cylindrical.
In detail the methods are quite unlike.

In order to compute (jfe), the thermal conductivity, the

input of energy into the section (MN) was equated to pjg 5

the radial flow across this section. This is expressed by

the equation

Pdl dT

J dx ^ ^

Pd is the potential drop across the specimen (length MN) ; /, the cur-
rent; /, the mechanical equivalent of heat; jfe, the thermal conductivity;
/, the length of the specimen (MN)(= 4.94 cm.); dT/dx^ the temperature
gradient at the distance x from the center; n, the radius of the hole
(= .536 cm.); r2, the outer radius of the oxide (—1.69 cm.).

»M. F. Angell, "Thermal Conductivity at High Temperatures," Phys. Rev., Vol.
XXXIII., No. 5, p. 421 (1910).

764 C. C. BI DWELL, ^SS

Equation (i) may be written

Pdl

n - r

2wkJl

and upon integration gives

*=2x//(rf-r.)'*'«-(7!)- ^"^

The table accompanying will serve to indicate the degree of accuracy of
the results. The average of the readings of A and B was taken as the
temperature of the inner surface; that of C and D, of the outer surface.
Even when very large temperature gradients were used, such as 133** in
one case, 76** in another, the values were found to be in good agreement
with results obtained with much smaller temperature gradients (40® or
thereabout). This is regarded as indicating the essential correctness
of the results. The justification for considering the average of the inner
junctions as giving the correct temperature of the inner surface, and the
average of the outer junctions as giving the correct temperature of the
outer surface is found in the general line-up of the points as shown on
the curve (Fig. 7). Unless the averages gave pretty closely the correct
values such consistent behavior would be hard to account for. Some
uncertainty necessarily exists concerning the exact composition of the
oxide at each temperature at which measurements were made. Chemical
analysis at the close of the work showed 70.0 per cent, iron, the exact
percentage required for FejOa. Since heating for some eight hours was
found necessary for steady conditions before thermal conductivity could
be measured, and since the first measurements were made at the upper
temperatures (1047**, 905° C, etc.) there can be very little doubt that
the oxide was at least very closely of the composition FejOs throughout
the observations. The indication of a linear relation between thermal
conductivity and temperature for the class of materials exemplified by
iron oxide is regarded as the significant part of this work. The data
were taken in the order given in the table.

The remarkable difference here brought out between the behavior of
electrical and thermal conductivity, the former increasing rapidly accord-
ing to an exponential law, the latter slowly according to a linear relation,
indicates probably a different mechanism involved in these two effects.
The behavior of electrical conductivity is explained by Konigsberger^
as due to the increase in the number of free electrons with rising tempera-
ture. An indication of the relative number of free electrons is given by
the thermionic emission from bodies of this class. For glowing lime

» J. KSnigsberger, Jahrbuch d. Radioaktivitat, Vol. II., p. 84, 1914.

No. 6. J

IRON OXIDE.

765

Data for Thermal Conductivity of Fe^i.

1.064'*
918
732

849H
861

880
895
308
282
439
536
616
677
645
668
729
735
750
777
855
193

1.072**

i,o;26*

1,024*

934

887

883

734

675

663

849H

798

792

863

812

806

881

831

825

898

847

842

310

173

175

281

217

216

441

388H

385 H

539

491

489

619

576

572

681

640

636

647

605

602

674

635

630

731

694

688

738

701

695

755

720

709

781

746

738

860

828

822

201

120

122

T«mp.
Diffennctt.

43*

52H
52

133

47H

43H
41

42^

39M
39

38H

32H
76

Temp.

1,047*

905
701
822
835
854
870
242
258
415
515
595
658
624
651
711
718
734
759
842
159

Enerry In-
put (Watts),

18.91
15.16
18.37
18.45
18.45
18.75
18.75
22.57
11.28
11.30
11.50
11.37
11.50
11.42
11.28
11.33
11.30
11.50
11.60
11.53
11.15

.00390
.00335
.00255
.00300
.00309
.00317
.00320
.00149
.00154
.00189
.00214
.00232
.00251
.00239
.00249
.00258
.00260
.00269
.00279
.00314
.00130

the law of electron emission is an exponential one of the same type as
that expressing the change of electrical conductivity with temperature.^
Thermal conductivity therefore seems to bear no simple relation to the
number of free electrons.

A very interesting phenomenon brought out by the curve is the break
at 720** C. corresponding very well with the break in the resistance and
thermo-electric power lines at that temperature.

Summary.

The work here reported is an extension of previous research on electrical
resistance and thermal electromotive forces in the oxides of iron. The
methods have been improved, the measurements of the two properties
have been made more nearly simultaneous and have been extended to
the melting point of the oxides (1520° C).

The chemical composition of the specimens has been investigated
and the difference in the electrical behavior of the two oxides, Fe208
and Fe804, shown.

A correction to a previous report has been indicated.

The previously reported transformation for FeaOs at yio^'-yso*' C. is
verified. The thermo-electric line below this point is again showh to be

> H. A. Wilson, Phil. Trans. A.. Vol. CCII., p. 243. 1903.

766

C. C. BIDWELL.

Straight, while above this point the more extended data indicate a very
different law, a maximum positive value being reached in the neighbor-
hood of 1 125® C, the values thereafter decreasing to zero and then
becoming n^ative, thus showing two reversals in sign. A transformation
at 1320** C. probably involving some change in structure is deariy indi-
cated by the behavior of the thermo-electric line in this r^on.
The electrical resistance in the case of FejOi is found to obey the

exponential law suggested
by Konigsberger for sub-
stances of this class and is
plotted to the correspond-
ing logarithmic equation.
When so plotted the two
transformations r e v e a led
by the thermo-electric pow-
er lines are strikingly cor-
roborated.

The electrical resistance
and thermo-electric power
of Fes04 as a function of
temperature is reported
upon. Owing to change in
composition (oxidation)
which occurs on heating,
the pure temperature varia-
tion is masked, but a wide
difference from the beha-
vior of Fe208 is shown.

Values of thermal con-
ductivity of FeiOi up to
1050** C. have been ob-
tained. The change of
thermal conductivity with temperature is found to be a linear one. The
data are shown graphically by two straight lines intersecting at approxi-
mately 720** C, the transformation point previous brought out by the
resistance and thermo-electric power lines.

The work was performed in large p)art in the summer of 1916 under a
grant from the Carnegie Institution of Washington.

Cornell University,
September, 191 7.

Fig. 7.
Thermal conductivity and temperature (FeiOi).

X^^f] ABSORPTION SPECTRA. 767

THE ULTRA-VIOLET AND VISIBLE ABSORPTION SPECTRA

OF PHENOLPHTHALEIN, PHENOLSULPHONPHTHALEIN,

AND SOME HALOGEN DERIVATIVES.

By H. E. Howb and K. S. Gibson.

TN this paper. are given the results of a quantitative study of the

^ absorption spectra of phenolphthalein and its halogen and sulphon

derivatives with particular reference to the effect of the substitution of

halogen for hydrogen and to the relation between chemical constitution

and absorption in cases where color results from the addition of alkali

to neutral solutions of the phthaleins.

The investigation was carried out in the physical laboratory of Cornell

University, having been made possible by a grant from the Carnegie

Institution to Professor E. L. Nichols. For information and suggestions

concerning the chemistry of the problem the authors have been dependent

upon Professor W. R. Orndorff and Dr. S. A. Mahood, of the department

of chemistry, who have supplied the substances studied and have followed

the work closely.

The Compounds Studied.

Phenolphthalein may be considered the parent substance from which
the other compounds studied are derived by the substitution of chlorine,
bromine, or iodine atoms for the hydrogen atoms attached to the carbon
atoms of the benzene rings of the molecule. This compound, which is
a condensation product of phthalic an- yv >v

hydride, OC— C6H4— CO with phenol, "Y| ^f" ""Q [V

CeHj.OH, is given the structural formula

shown in Fig. i (a). Phenol tetrachloro-

phthalein is a derivative in which four

chlorine atoms are substituted for the

hydrogen atoms in the phthalic acid res- ^*^* ^*

idue, i. e., for the four hydrogen atoms ^^^ Phenolphthalein.

, . , • 1 « fl . . (fr) Di-potassium salt of phenolphtha-

which are in the lower benzene ring in i^i^
the formula.

Both of the above compounds may have derivatives formed by the
substitution of two bromine atoms or two iodine atoms in each of the

768 H, E, HOWE AND K, 5. GIBSON, [

benzene rings of the phenol part of the molecule. The resulting

compounds are named tetrabromo- and tetraiodophenolphthalein and

-phenoltetrachlorophthalein. The six substances are colorless in the

finely divided crystalline form in which they are obtained. They do not

dissolve in alkali-free water, but dissolve in neutral alcohol, forming

colorless solutions. The addition of alkali to the neutral solutions causes

the appearance of color.

The appearance of color in such cases has generally been assumed

to be "accompanied by the transformation into a derivative of quinone," ^

and the graphic formula given to the di-potassium salt is that shown in

Fig. I (ft).

Apparatus and Procedure.

The absorption in the visible part of the spectrum was measured with
a Lummer-Brodhun spectrophotometer,* with the acetylene flame as a
source. The ultraviolet absorption was determined photographically
by means of a Hilger sector photometer* in connection with a large Hilger
quartz spectrograph. The source in this case was the aluminum spark
under water, which gives a continuous spectrum as far as the quartz
system will transmit.

The absorption curves are plotted to show the molecular absorption
constant as a function of the frequency. The frequency is the reciprocal
of the wave-length in millimeters, e, g,, wave-length 5,000 A.U. equals
frequency 2,000. The absorption constant j8 is defined by the equations

J, = lo^** or )8 = ^ X Logio J, ,

where / represents the intensity of light transmitted by a cell filled with
the pure solvent, /' the intensity' of light transmitted by a similar cell
filled with the solution, d the thickness of the absorbing layer in centi-
meters, and c the concentration of the solution. As a concentration of
.0001 gram-molecule per liter, i. e., .0001 iV, was found convenient for
the photographic work, this concentration was taken as. the unit. If the
concentration were expressed in gram-molecules per liter, the constant j8
obtained would be 10,000 times that plotted. While a concentration of
.0001 iV and a thickness of i cm. was satisfactory for most of the photo-
graphic work, the weak color of certain solutions made it necessary to
vary the concentrations from i^/ioooo to N/64, ^^^ the thickness of the
absorbing layer from .3 cm. to 10 cm.

A check on the accuracy of the measurements is given by the over-

* Perkin and Kipping. "Organic Chemistry," p. 531.

* K. S. Gibson, Physical Review, 7, p. 194, 1916.

* H. E. Howe, Physical Review, 8, p. 674, 1916.

vol.x.^

Na6. J

ABSORPTION SPECTRA.

769

lapping of the curves obtained visually and photographically. The visual
readings were extended in the blue nearly to frequency 2,200, and the
photographic measurements could be made to a frequency approximately
2,000. The agreement of the two methods was very good in most cases.

The solutions were prepared in the following manner. A weighed
amount of phthalein was dissolved in neutral absolute alcohol, the
amount of alcohol being so chosen that a stock solution was obtained
somewhat stronger than was desired for study. Neutral solutions for
study were made by further diluting this stock solution, while alkaline
solutions were prepared by adding to a measured volume of the stock
solution a calculated volume of N/ioo solution of potassium hydroxide
in absolute alcohol until there were present one, two, four, or ten mole-
cules of alkali for each molecule of phthalein, after which more alcohol '
was added to obtain the desired concentration of phthalein.

Aque')us solutions cannot be so prepared, as the phthaleins will not
dissolve in pure water. A weighed amount of phthalein was dissolved
in a calculated volume of N/ioo potassium hydroxide in water, and more
water was added to obtain the desired concentration of phthalein.

Details of Absorption Spectra.

I. The Effect of the SubsHtution of Halogens.

The neutral .0001 iV alcoholic solutions of the six phthaleins studied
give absorption curves shown in Fig. 2. All six curves show more or less
plainly two bands near frequency 3,500, and increasing general absorption

Neutral Alc»h0ht Safutitm*

4000

beyond. 3,800. Weaker solutions showed no other bands with frequencies
less than 4,200, the working limit of the apparatus.

The short vertical lines indicate the positions of band centers as
estimated directly from the negatives. When a negative is viewed as a
whole the contrast effects often make the bands seem plainer than the

770

H. E, HOWE AND K, 5. GIBSON,

curves show them, since the points plotted are found by examining a
very small portion of the plate at a time. For example, the tetraiodo-
phenoltetrachlorophthalein n^^tive showed indications of two bands
when viewed as a whole, while the actual values obtained gave a curve
with merely a broad shoulder. The probable shape of the curve is
indicated by a dotted line. The use of plate contrast to locate bands
would not be allowable if the source gave other than a continuous
spectrum.

The substitution of bromine or iodine for hydrogen increases the
absorption and shifts the bands of lower frequencies, apparently
without any change of frequency difference. The presence of four
chlorine atoms in the phthalic acid residue, besides adding a shoulder
to the side of the curve, seems to be accompanied by a coming together
of the two bands. As seen on the curves, this approach seems to be
effected by the shift of the band of lower frequency toward higher
frequencies, and it is interesting to note that for alkaline solutions a
similar shift of the ultra-violet bands toward higher frequencies accom-
panies the substitution of chlorine (Table I.).

Table I.

Frequencies of Absorption Bands for Alkaline Solutions,

Alcoholic Solutions.

Phenolphthalein

Tetrabromophenolphthalein

Tetraiodophenolphthalein

Phenoltetrachlorophthalein

Tetrabromophenoltetrachlorophthalein

Tetraiodophenoltetrachlorophthalein

Aqueous Solutions.

Phenolphthalein

Tetrabromophenolphthalein

Tetraiodophenolphthalein

Phenoltetrachlorophthalein

Tetrabromophenoltetrachlorophthalein

Tetraiodophenoltetrachlorophthalein

Band Proquencies.

1,780
1.700
1.675
1.715
1,625
1.600

2,7501

2.7501

• • • •

3.210
3.195
3.570
3.265
3.250

1.810

2.710

1,725

2.550»

1,685

2.500»

1.740

2.760

1.640

2.650*

1,610

2.570

3,500*

3,280

3.200

3.590

3,290

3,240

The change from neutral to alkaline solution is accompanied by a
marked change in the character of the absorption (Figs. 3 and 4), which
now shows a band in the visible and a single band near 3,200, followed
by increasing general absorption beyond 3,500. The following deviations

1 Frequency of band center only approximate.

Vol. X.l
No.6. J

ABSORPTION SPECTRA,

771

from this general result may be cited : phenolphthalein with ten molecules
of potassium hydroxide shows a weak band near 2,700 and a very slight

i»

A .—iN. Nemtrml
r MtH «• • •

Z$«0

A .MM At Mt»rr0t
B »»«tH '

P S«SfM. 4 ' ' •

Ak^

'fX.

r 00, N m '

C AMf/V 1 •

1 ^

t»»9 Ilk

J«0«

4#^ '

^ooo

sooo

Fig. 3.

indication of a band near 3,400, while phenoltetrachlorophthalein with
six or ten molecules of alkali shows bands near 2,700 and 3,500. For
the exact location of the bands, see Table I.

1990

nrrmi***ith»0ftrtrrt*M4ft^thahim

B .#M/M iHtl. KOM

_P .#••/ N t * •
Im Ak«A»t

ecoo

Fig. 4.

772

. £. aOWE AND K. S. GIBSOK.

Both visible and ultra-violet absorptions increase with the amount of
alkaU present up to twenty molecules, though this increase is not so
rapid after the total amount of potassium hydroxide is more than two
molecules per molecule of phthalein.

The band in the \~isible shifts toward the red with the substitution of
chlorine, bromine, or iodine for hydrc^;en (Fig. 8).

The dilute solutions of tetrabromophenol phthalein and tetraiodo-
phenolphthalein used for photographing did not show color through a
I cm. layer. In order to obtain the curves of absorption in the visible
r^on it was necessary to use stronger solutions. It may be noted that
the ultra-violet absorption of the apparently colorless solutions is of the
same type as that of the colored solutions of the related fx)mpounds.

.-.

i '

• -■

,*'^*T

_,^

Ai-/

1 1

'iv

FiK. 5.

Fig. 6.

In the visible region the absorption bands of the aqueous solutions
(F<Ks. 5 and 6) are similar to those of the alcoholic solutions but the value
of the maximum absorption is increased two to nine times and the bands
are shifted toward higher frequencies. The solutions show increasing
absorptions in the same order as do the alcoholic solutions of the same
strength of alkali, viz., tetrabromophenotphthalein, tetraiodophenol-
phthalein, tetrabromophenol tetrachlorophthalein, tetraiodophenoltet-
rachlorophthalein, phenolphthalein, and phenoltetrachlorophthalein.
The last-named substance has a maximum absorption i8o times the first
in aqueous solution and 450 times in alcoholic solution.

AH of the aqueous solutions show weak ultra-violet bands near fre>

IL. X.l
>. 6. J

VOL.X.1

No.

ABSORPTION SPECTRA.

773

jOfi^

^ou

MOU

quency 2,600. To bring out this band photographs were made with
stronger solutions and the points so obtained are indicated on the curves
by small circles close together. The solutions of tetrabromo- and
tetraiodophenolphthalein containing potassium hydroxide did not show
this band with the low concentrations first used. As such a band had
been earlier reported by Meyer and Fischer,^ solutions were prepared
by their method of treating an excess of phthalein with half normal
sodium hydroxide, filtering off the undissolved phthalein, and diluting
the solution to the concentration indi-
cated in Fig. 7, which shows the band
in question. The value of the absorp-
tion constant here shown is not strictly mb
comparable with that plotted in the
other cases because of the different
method of preparing the solution.

The maximum absorption in this weak
band changes with the phthalein in the
same order as does the band in the
visible. This indicates a connection be-
tween the bands and suggests that a

corresponding band in the ultra-violet should appear in the alkaline
alcoholic solutions. Such a band does appear in the solutions that are
most strongly absorbing, t. e,, phenolphthalein and phenol tetrachloro-
phthalein.*

This band, like the band in the visible, shifts toward lower frequencies
with the substitution of bromine or iodine. But unlike the band in the
visible, it shifts in the opposite direction upon the substitution of chlorine.
A shift of this latter sort is shown by the stronger ultra-violet band near
3,200 when chlorine is substituted. This is true in both alcoholic and
aqueous solutions. Hence in its shifts the weak ultra-violet band seems
related to the other band in the ultra-violet, while in its variations in
intensity it seems related to the band in the visible.

In striking contrast with the behavior of the bands in the visible and
near 2,600, the prominent band near 3,200 in the ultra-violet shows a
maximum absorption of approximately the same value for all solutions,
alcoholic and aqueous. The absorption is slightly increased by the sub-
stitution of chlorine.

Fig. 8 shows the collected curves of visible absorption of the alkaline

* Ber. d. Deut. Chera. Gesell., 44, p. 1944, 191 1.

* Since this paper was completed a careful examination of alkaline alcoholic solutions of
tetrabromo- and tetraiodo-phenoltetrachlorophthalein of greatly varying concentrations
has been made. No trace of the band near frequency 3,600 was found.

H. E. HOWE AND K. S. GIBSOlf.

[

Ailuuona, The solutions used for obtaining the curves in the upper
ii>w contained ten molecules of potassium hydroxide per molecule of
l>hthalcin. Tetrabromo- and tetraiodophenolphthalein solutions were
•*u weakly colored that the only solutions examined in the visible were
A'/iooo with ten molecules of alkali. The other solutions whose absorp-
tions are shown in Fig. 8 con-
tained four molecules of alkali.
While the shifts toward the
red in the visible region are
seen from Table I. and Fig. 8
to follow the general law of
increase with increasing mass
of substituent, it is also to be
seen that the frequency of
band center is not a function
of molecular weight alone.
Thus the salt of tetraiodophe-
nolphthalein (mol. wt. 898) has
a band centering at a frequency
higher than that of tetrabro-
mopheno Itetrach lorophthalein
(mol. wt. 848). The position
of the absorption band might
reasonably be expected to de-
pend upon both the halogen
substituted and upon its posi-
tion in the molecule. With
this in mind, the frequencies
of band centers were plotted
against molecular weights (Fig.
9) when it was found that the
points lay on two curves cor-
responding to the two groups
into which the compounds may
be divided. One group includes
phenolphthalein and those of its derivatives in which bromine or iodine
are substituted in the phenol part of the molecule. The other group
contains phenoltetrachlorophthalein and its corresponding derivatives.
The authors propose to predict from these curves the centers of absorp-
tion bands that would be found for solutions of phthaleins in which
chlorine is substituted in the phenol p)art of the molecule or in which
bromine or iodine is substituted in the phthalic acid residue.^

Fig. 8.

Voi.X.1
Na6. J

ABSORPTION SPECTRA,

775

Table II.

Alkaline Aqueous Solutions.

Phenolphthalein .

Tetrabromophenolphthalein .

Band No.

Previously Ponnd.

I /A.

2
3

.5585'

.5500"

.372«

.581»

.5835«

.389»

1.790
1,820

2,690

1,720
1,715
2,570

Pound by Author*.

I/A.

1,810

2,710

3,500 (approx.)

1,725

2,550 (approx.)

3,285

> Meyer and Marx, Her. d. Deut. Chem. Gesell., 41. p. 2446. 1908.
* Meyer and Fischer, Her. d. Deut. Chem. Gesell., 44> P- I944* 191 1*

tei9

^ Jitrai»^»ph000ltttra9Mhn^tMtiii

Table II. gives a comparison
of the frequencies of the absorp-
tion bands for alkaline aqueous
solution s as taken from the
curves prevously published and
as found by the authors. The
failure of Meyer and Fischer to
detect band No. 3 was probably
due, in the case of phenolphtha-
lein, to the fact that the band is
a broad faint one superposed
upon the increasing general ab-
sorption, and in the case of tetra-
bromophenolphthalein, to their
failure to carry the concentra-
tions over a great enough range.
In this latter substance, band

No. 2 was found by the authors "' ^•'- ^^'

not to be so pronounced (Fig. 7) F'ig- ^^

as the curve given by Meyer and Fischer would indicate.

1 At the time this prediction was proposed, phthaleins with the suggested halogen substitu-
tions had not been prepared. Since this paper was completed, tetrachlorophenolphthalein
has been prepared under the direction of Dr. S. A. Mahood. of the Department of Chemistry,
and its absorption band determined by Mr. E. P. Tyndall. who found that alkaline solutions
gave bands centering at 1695 in alcoholic solution and at 1,720 in aqueous solution. (See
the following article.) Since the molecular weight of the di-potassium salt of the compound
is 532, the frequencies predicted from the curves are i,739 and i.770t and are far from
agreement with those actually found. A further test of the prediction would be interesting.

776 H. E, HOWE AND K. 5. GIBSON. [

2. The Relation of Absorption to Constitution,

The difference in type of absorption spectra of solutions of a phthalein
and of its di-potassium salt may be taken to mean that there is a funda-
mental difference in the structure of the two molecules. One way of
representing this structural difference is shown in the graphic formulas
given in Fig. i for phenolphthalein.

If all the alkali added to a solution of the phthalein reacted to form
the colored salt, the conversion of the phthalein would be complete
when there had been added two molecules of potassium hydroxide per
molecule of phthalein, and the addition of more alkali would produce no
effect on the absorption. The curves for the alcoholic solutions show
that such a complete conversion of the phthalein does not take place.
For example, the visible absorption of phenolphthalein could not be
measured when only two molecules of alkali were present, and it con-
tinued to increase with alkali up to the greatest amount added, i. e.,
ten molecules. Further, when colored alcoholic solutions of phenol-
phthalein and of phenoltetrachlorophthalein containing two molecules
of alkali were diluted with alcohol the color disappeared and the absorp-
tion in the ultra-violet reverted to the neutral type. This means that
the excess of alcohol decomposes the salt, setting the phthalein free.
The potassium ethylate resulting from the combination of the alcohol
with the potassium of the colored salt has no absorption. Hence the
solution gives the same absorption as the neutral solution of the phthalein.

The absorptions of solutions containing different relative amounts of
phthalein and alkali were measured for the purpose of obtaining informa-
tion concerning the compounds formed in the different cases. The
curves of ultra-violet absorption of solutions with one molecule of alkali
(Figs. 3 and 4) show that a partial conversion of the phthalein has taken
place. The solutions showed a very slight color, with one molecule of
alkali. In the visible region the absorption merely increases without
change of type after the first appearance of color.

However, with a related compound, viz., phenolsulphonphthalein, it
was possible to follow the gradual transition in type of spectrum in both
visible and ultra-violet as the amount of alkali was increased. Fig. 10
gives curves for the neutral solution and for solutions containing i, iH»
and 2 molecules of alkali. The graphic formula is given, showing the
quinoid structure assigned to this compound because it is colored.
The neutral solution is yellow, the alkaline solution red. The change
in color is progressive, due to the growth of a band in the yellow-green.
The absorption of the alkaline aqueous solution (bands centering at
1 1785, 2,770, and 3,500) differs greatly in type from that of the neutral
solution (bands at 2,320 and 3,770).

VouX.1
No. 6. J

ABSORPTION SPECTRA.

777

Fig. II shows a similar change for alcoholic solutions. A rough
examination of the absorption of tetrabromophenolsulphonphthalein
indicated similar changes also. The physical evidence for a change in
structure is as good in the case of these substances as in that of the

Tfn ^ Tfgr

' Fig. 10.

ft •> • mll0l.KOH

4$§¥

Other phenolphthaleins discussed. The bearing of the facts here pre-
sented on chemical theory will be discussed in a later paper by Prof.
W. R. OrndorflF. It is hoped to continue the collection of data on absorp-
tion and constitution.

Summary.

From a study of Table I. and the curves the following summarized
statement of facts can be made.

Neutral alcoholic solutions of the phthaleins studied have absorption
spectra of the same type (Fig. 2).

The type of absorption changes when the solutions become alkaline
(Figs. 3, 4). In some cases the change can be followed through a transi-
tion stage.

778 H. E. HOWE AND K. S. GIBSON, [sSS

The absorption of the aqueous solutions is of the same type as that
of the alcoholic (Figs. 5, 6). The characteristic spectrum of the alkaline
solutions consists of three absorption bands, one in the visible in the
region 1,600-1,800, the other two in the ultra-violet in the regions
2,500-2,700 and 3,200-3,600.

The maximum value of the absorption constant in the visible and in
the band near 2,600 varies greatly with the phthalein, and is considerably
greater in aqueous than in alcoholic solutions.

Band centers have a lower frequency for alcoholic solutions than for
aqueous (Figs. 3. 4» 5. 6» S)-

The band in the visible region is shifted toward lower frequencies by
the substitution of bromine or iodine in the phenol part of the molecule
and of chlorine in the phthalic acid part of the molecule. The shift
increases with the mass of the substituent and is less in alcohol than in
water. Chlorine adds a shoulder to this band on the side toward higher
frequencies (Fig. 8).

The band near 2,600 is shifted in the same direction as is the visible
band by bromine and iodine, but is shifted toward higher frequencies
by the chlorine, the shifts increasing with the mass of the substituent.

The band near 3,200 is also shifted toward lower frequencies by
bromine and iodine and toward higher frequencies by chlorine. As in
the case of the other bands, the shift by iodine is greater than by bromine.

It is hoped that the accumulation of further data may make possible
general conclusions as to the relation of absorption to constitution of
the phthaleins.

Cornell University,
August, 191 7*

No*^] ABSORPTION OP TETRACHLOROPHENOLPHTHALEIN.

779

NOTE ON THE ABSORPTION OF TETRACHLOROPHENOL-

PHTHALEIN.i

By R. C. Gibbs, H. E. Howe, and E. P. T. Tyndall.

SINCE the work reported in the preceding article was completed, a
new derivative of phenolphthalein, tetrachlorophenolphthalein, has
been prepared under the direction of Dr. S. A. Mahood, of the department
of chemistry. Its ultra-violet and visible absorption has been measured.
The curves in Fig. i show the type of absorption of solutions of this
compound. The frequencies of band centers and maximum coefficients
of absorption are tabulated below.

Solution.

i/X.

i/x.

Neutral alcoholic

1.695
1.720

.0025
.0203

3,435
3.535
3,270
3,300

.59

Alkaline alcoholic

.64
1.17

Alkaline aqueous

1.05

In the ultra-violet the absorption of tetrachlorophenolphthalein closely
resembles that of tetrabromophenolphthalein. In the neutral alcoholic
solution the band centers and their absorption coefficients lie between
those for phenolphthalein and tetrabromophenolphthalein (see Curves
A and B, Fig. 2, of preceding article). When alkali has been added the
solution shows the band characteristic of this series of compounds.
The center of this band lies at 3,270, a frequency higher than that for
tetrabromophenolphthalein, the compound of next greater molecular
weight.

Since in tetrachlorophenolphthalein, the chlorine is substituted in the
phenol part of the phenolphthalein molecule, it would be expected that
this substance would be similar in its physical properties to the corre-
sponding tetrabromo and tetraiodo derivatives. Its ultra-violet absorp-
tion indicates such similarity. The center of the visible band would be
expected to lie between the centers of the bands for phenolphthalein
and tetrabromophenolphthalein. On the assumption that molecular
weight is the determining factor the authors of the preceding article

^ The investigation deacribed in this report was carried on with aid from the Romford Fund
of the American Academy of Arts and Sciences.

78o

R. C. GIBBS, a. B. HOWE AND E. P. T. TYNDALL.

rSkOOMD

LSbub.

predicted from their Fig. 9 that the band center would lie at 1,739 in
alcohol and at 1,770 in water. From the table of determined values it
will be seen that the band centers not only do not lie where predicted
but are at even lower frequencies than for the heavier molecule of tetra-
bromophenolphthalein. The differences between the absorption of solu-
tions of tetrachlorophenolphthalein (Fig. i) and of solutions of phenol-
tetrachlorophthalein (Figs. 2 and 4 in the preceding article) indicate

AMmUHmml BmnMNutmt AUidc
Cm^tHifhiKOH thmN*MndKOH'

Fig. 1.

that the position of a substituent in the molecule is probably a factor in
determining the nature of the absorption.

The bands in the aqueous solution come at a higher frequency than
those in the alcoholic solution as was found for other members of the
series.

Several compounds in this series gave absorption bands near 2,600
(Figs. 3, 4, 5, and 6 of the preceding article). The examination of
solutions of various concentrations brought out no evidence of such a
band in tetrachlorophenolphthalein.

It was desired to determine whether the absorption of an alkaline
alcoholic solution of a phthalein prepared by adding potassium ethylate
to a neutral solution would differ from that of a solution to which the
alkali had been added in the form of a hydroxide. A solution of potas-
sium ethylate was prepared by dissolving metallic potassium in absolute
alcohol, and this solution was used in the same way as the solution of
potassium hydroxide in alcohol to make up solutions containing any

Na*6^] ABSORPTION OP TETRACHLOROPHENOLPHTHALEIN, 78 1

desired number of molecules of alkali per molecule of phthalein. Solu-
tions of the strength cited below were examined using both forms of
alkali. No difference greater than that due to observational errors could
be detected in any two solutions containing the same amount of alkali
in different forms.

Visible Absorption.

Tetrachlorophenolphthalein 0022 AT, 10 KOCtH* and .002 N, 10 KOH.

Tetraiodophenoltetrachlorophthalein 0005 N, 1 KOCsHb and .0001 N,^ 1 KOH.

0001 iVr. 2 KOCjH* and .0001 iVr.i 2 KOH.

UUrO'VioUi Absorption,
Tetrachlorophenolphthalein 0001 iVT, 1 KOCtH* and .0001 N, 1 KOH.

0001 N, 4 KOCjH* and .0001 iVT. 4 KOH.

Tetraiodophenoltetrachlorophthalein, 0001 N, 1 KOCsH» and .0001 N,^ 1 KOH.

0001 AT, 2 KOCjH* and .0001 N^ 2 KOH.

^ Solutions examined by Howe and Gibson. See preceding article.

Cornell University,
Ithaca, N. Y.

782 MBCE NAD SAHA.

ON THE LIMIT OF INTERFERENCE IN THE FABRY-PEROT

INTERFEROMETER.

By Mech Nad Saha.

WHEN a monochromatic source of radiation {ior example that
given by a vacuum tube, when excited by an dectric discharge)
is examined by a Fabry-Perot interferometer, we obtain bright and
narrow rings of maximum intensity separated by wide daric intervals.
If the distance between the plates of the 6talon be gradually increased,
the maxima gradually decrease in brightness, until we reach a limit
where we can no longer distinguish between the maxima and the minima.
The theory of this phenomenon has been worked out by Lippich, Lord
Rayleigh,^ and SchSnrock,* and is shown to be due to the fact that the
emission centers (in this case the gaseous atoms) being in motion, a sort
of D6ppler-Fizeau effect is produced in the line of vision of the observer.
They have shown that when the pressure is small, the critical distance D
(or the limit of interference) is connected by the following formula
with the wave-length (X) of light, the temperature (J) of the tube,
and the mass {M) of the emission centers:

D \m

(a)

This theorem has been made the basis of a wide series of experiments
by Michelson,* and the French School of opticians including Fabry,
Perot, and Buisson.* Among the various problems to which the formula
(a) has been applied may be mentioned the following:

(i) The temperature of the discharge tube when emitting a nK>no-
chromatic light.

(ii) The temperature of stars and nebulae.

(iii) Mass of the emission centers of lines in the spectrum. Probably
the mass of the emission centers of many lines of unknown origin in the
solar corona and many nebulae {e, g., \ — 5007 A. U.) which are at-

* Lord Rayleigh, Phil. Mag., November, 1915.

' Schdnrock, Ann. d. Physik., 1907, Bd. 23, 1907.

» Michelson, Astro-physical Journal, 1895. Vol. (ii), p. 251.

^ Buisson et Fabry, Journal de Physique, tome 11, 1912, p. 442-464.

Na*^] PABRY-PEROT INTERPEROMETER. 783

tributed to h3rpothetical elements^ coronium and nebulium may be deter-
mined by this method.

The value of the constant il is of much use in all these investigations,
and it is generally deduced from theoretical considerations. While
going through the literature on the subject, I found that A is generally
calculated from approximate and not altogether satisfactory considera-
tions, though an exact solution is not difficult. My object in the present
communication is to effect this improvement in the theory. For this,
we must begin with a preliminary scrutiny of the theory of the Fabry-
Perot interferometer.

The Fabry-Perot interferometer consists of two plane parallel plates
of glass, both heavily silvered on the inside. If a ray of light is sent
through the plates, it undergoes several internal reflections, and at each
reflection from either surface, a part issues out. Every incident ray is
thus subdivided into a large number of parallel rays. If the angle on
incidence is very small, almost normal, as is the case in practice, the
number would be infinite. Let us confine our attention to the rays
issuing on the side further from the source of light. The parallel rays
issuing at some particular angle have path differences amounting to
2d cos a, 4d cos a, 6d cos a, etc., according as they have suffered
double reflection once, twice, thrice, or any number of times. When
these rays are brought together by a converging lens we shall have the
interference phenomena. The parallel system is composed of rays
transmitted directly, t. e., without reflection — this ray can be represented
by £0 cosn/; rays suffering reflection twice, four times, etc. Since at
each double reflection there is a retardation in phase amounting to
2tA/X and the intensity is reduced by a fraction /, we can represent the
rays by the equations

where we put

2tA
A = 2i cos a, and d = —r- •

The resultant ray is now represented by

E = £o{cos nt+f cos (nt - d) + f cos (nt - 2d) + • • • }

= £o[cos nt{i +fcos 6 +P cos 2d + • • • }

+ sin nt{f sin 3 + /* sin 25 + • • • }]

^ T I — fcos S , . fsind 1

= £0 I cos n/ • -z > , ^ + sm nt • 7 > , ., I .

L 1 — 2fcosS +P I — 2/ cos 5 + /* J

^ Nicholson, Phil. Mag., 191 1, Vol. 22, p. 864.

784 iiECH NAD SAHA [i

Therefore the intensity

/ = /,

i-2/cos5+/^ (I-/)* 4/ . . ^

This is the ordinary theory of the interferometer. The intensities of

the maxima and the minima are all in the ratio of i :

.+ -^

If we take / = .75, this ratio becomes 49 : i, the angular separation
being a = X/A. If the theory held rigorously, we could observe inter-
ference with large values of A. But this is not the case. For example
in the case of the sodium Z)i-line, no interference can be obtained when
A exceeds 3 cm. This is due to the fact that the radiant particles are
themselves in motion, and the theory cannot be perfect unless we take
account of this fact.

According to Maxwell's distribution law, the number of particles
having their velocity between V and V + dV is Ae^^^dV. The fre-
quency of radiation emitted by these particles is n[i + (v/c)] where n
is the wave frequency of light emitted by particles at rest. In the
expression for retardation in phase, we must therefore replace X by
X/(i + (v/c)] and write 2tA/X[i + (v/c)] in place of 2irAfk.

The intensity of light emitted by molecules having their velocity
between V + J V and V is

"^^ " ^ I - 2/ cos 8[i + (v/c)] +P'

The total intensity

e-^'^'dV

^bT —

2fcos8[i + (v/c)]+P'

We have by trigonometry,

I -P

z r~r~r^ = i + 2/ cos 5 + 2/ * cos 25 -h • • •

I — 2/ cos 6 + /* -^ ''

Now, we have

c/— 00

.^Ft,.. . W^V

dV'sin — = o,
c

£.-.".. rco,f-^«-

(llfiXf^lo)^

We have therefore

/ = ^"1772 Jj[^ +2£/»co8n«J.

Nolr^i PABRY'PEROT INTERFEROMETER. 785

Now let /i = the maximum value of /, corresponding to nd = o,
It = the minimum value of J, corresponding to nd = x.
Then the visibility factor V is, according to Michelson

__ 1 1 — U _ fe '^ +/'g -r * • •

Now (i/p)(2irAfKcy is of the order lo*. We can, therefore, safely omit
terms containing /*,/*, etc.

7 is therefore = 2fr^"^^^^'^''^\
From the kinetic theory of gases, we have fi = (m/2KT) = {<aM/2KT)^
where in = weight of the radiant atom in grams,
cu = weight of the hydrogen atom,

M = atomic weight of the radiant gas,

K » universal gas constant,

T = temperature.
Then we have, since

I /2tA\« , iV\ Ac \<aM^ i2f\

C6>

Lord Rayleigh took account of the first two interfering beams only,
but by this he had evidently the Michelson interferometer in his mind.
But I think that when we are applying the result to the Fabry-Perot
interferometer, we should take into account all the infinite number of
interfering beams, and the effect of reflection. This is exactly what has
been done in the present communication.

The exact evaluation of the constant ""-\l~^lo8^«l'rr) » cannot be

done unless the reflecting power of the plates, and the value of V be
known. / will depend upon the silvering of the plates, while V will
vary with the observer. Thus Lord Rayleigh takes the visibility factor
equivalent to .025, while SchSnrock takes it equivalent to .05. Assuming
that V = .025, and/ = .75.
We have

- = 1.50 X io«

ST'

While according to Lord Rayleigh

A . \M

- = 1.42 X I0« ^ y.

786 MBGH NAD SAHA.

As it is, the discrepancy between the two values if calculated by two
different methods is not much. But for particular apparatus, and for
particular observers, the discrepancy may be considerable. It is to be
hoped that investigators will take notice of these facts.

Calcutta University Collbgb op Scismcb,
July 7. 1917.

Vot.X.
Na6.

] BRItATA. 787

ERRATA.

Vol. IX., July, 1917, page 21, article by S. J. Bamett, entitied "The
Magnetization of Iron, Nickeli and Cobalt by Rotation and the Nature of
the Magnetic Molecule**; in last line of footnote, for "reached" read
"marked.**

Vol. IX., August, 1917, page 213, article by I. G. Priest, entitled "A
Proposed Method for the Photometry of Lights of Different Colors"; in
line 30, column 2, "5185** should read "0185.**

Vol. IX., October, 191 7, page 355, article by H. L. Howes and D. T.
Wilbur, entitled "The Fluorescence of Four Double Nitrates*'; in the
caption of Fig. 2, for "potassium** read "ammonium,** for "ammonium**
read "potassium."

788

INDEX TO VOLUME X.

Index to Volume X., Series II.

Abflorption, The Critical, of Some of the
Chemical Elements for High Fre-
quency X-Rays, F. C. Blake and
WiUiam Duane, 697.

Abflorption, Note on the, of Tetrachloro-
phenol-phthelein. R, C. Cibbs, H. E.
Howe and E. P, TyndaU, 779.

Absorption Bands, The High Frequency, of
Some of the Elements, 98.

Absorption Spectra, On the occurrence of
Harmonics in the Infra-red, of Gases,
W. W, CobUnis, 96.

Actinium Emanation, The Diffusion of, and
the Range of Recoil from it, L. W.
McKeehaHt 473.

Aeroplane, The Motion of an, in Gusts, E. B.
Wilson, 89.

Alpha-Radiation, A Reactive Modification
of Hydrogen Produced by, WiUiam
Duane and Gerald L. Wendt, 116.

American Phsrsical Society:

Abstracts, 74* i94* 589*
Minutes, 72.

Arnold, H. D., The Thermophone as a Pre-
cision Source of Sound, 22.

Atoms, Elastic Impact of Electrons with
Helium, J, if. Benade, 77.

Atomic Structure, Radiation, and R. A,
MiUikan, 194.

Audion-Tjrpe Rjadio Receivers, Internal Re-
lations in, Ralph Bown, 253.

Baker, W. C, A Single Construction for a
Condensation Pump, 642.

Bamett, S. J., The Magnetization of Iron,
Nickel, and Cobalt by Rotation and
the Nature of the Magnetic Molecule,

Bates, Frederick, Natural and Magnetic
Rotation at High Temperatures, 90.

Benade, J. M., Elastic Impact of Electrons
with Helium Atoms, 77.

Bichowsky, Russell v.. The Necessary Physi-
cal Assumptions Underlying a Proof
of Planck's Radiation Law, 92.

Bidwell, C. C, Electrical and Thermal Prop-
erites of Iron Oxide, 756.

Binary Alloys, Optical Constants of the. of
Silver with Copper and Platinum,
Louis K. Oppitt, 156.

Birge, Raymond J., A New Theory Con-
cerning the Mathematical Structure
of Band Series. 88.

Bishop, F. M., The Ionization Potential of
Electrodes in Various Gases, 244.

Blake. F. C, The Measurement of "A" by
Means of X-Ra3rs, 93.
The High Frequency Absorption Bands

of Some of the Elements, 98.
The Value of **k'* as Determined by

Means of X-Ra3rs, 624.
The Critical Absorption of Some of the
Chemical Elements for High Fre-
quency X-Rays, 697.

Booth, Harry T., Distribution of Potential
in a Corona Tube, 266.

Bown, Ralph, Internal Relations in Audion-
Type Radio Receivers, 253.

Brainin, C. S., An Experimental Investiga-
tion of the Total Emission of X-Rayt
from Certain Metals, 461.

Cady. F. B., Color Temperature Scales for
Tungsten and Carbon, 395*

Calcium, TheSpedficResistance and Thermo-
electric Power of Metallic. Charles
Lee Swisher, 6oz.

Carbon, Color Temperature Scales for
Tungsten and. E. P. Hyde, F. E.
Cady and W, E. Porsythe, 395.

Carson, John R., On a General Expansion
Theorem for the Transient Osdlla-
tions of a Connected System, 217.

Cheney, W. L., The Emission of Electrons
by a Metal when Bombarded by Posi-
tive Ions in a Vacuum, 335.

Coblentz, W. W., On the Occurrence of Har-
monics in the Infra-Red Absorption
Spectra of Gases, 96.
The Use of a Thomson Galvanometer
with a Photoelectric Cell, 97.

Color Temperature Scales for Tungsten and
Carbon, £. P, Hyde, F. E, Cady and
W, E. Porsythe, 395.

Compton, A. H., The Reflection Coefficient
of Monochromatic X-Rays from
Rock Salt and Caldte, 95.

Compton, K. T., Theory of Ionization by
Partially Elastic Collisions, 80.

Condensation Pump, A Single Construction
for a, W, C. Baker, 642.

Cooksey, C. D., The K Series of the X-Ray
Spectrum of Gallium, 645.

Coordinates, Generalized, Relativity and
Gravitation, E, B, Wilson, 89.

Corona. The Pressure Increase in the, EarU
H. Warner, 483.

VOL.X.1

No. 6. J

INDEX TO VOLUME X.

789

Corona Tube, Distribution of Potential in a,
Harry T, Booth, 266.

Crandall, I. B., The Thermophone as a Pre-
cision Source of Sound, 22.
The Composition of Speech, 74.

Crawford. William W., The ParaUel Jet High
Vacuum Pump, 557.

Crebore, Albert C. Theory of Crystal Struc-
ture, with Application to twenty Crys-
tals belonging to the Cubic or Isomet-
tric System, 432.

Crjrstal Structure, Theory of. with Applica-
tion to Twenty Crjrstals belonging to
the Cubic or Isometric System, Albert
C. Cr chore, 432.

Davis, Bergen, Ionization and Excitation of

Radiation by Electron Impact in

Mercury Vapor and Hydrogen, loi.
Demagnetization of Iron, Arthur Whitmore

Smith, 284.
Density, A Determination of the, of Helium

by Means of a Quartz Micro-Balance,

r. S. Taylor, 653.
Dixon, A. A.. The Ionizing Potentials of

Gases, 495*
Doubt, Thomas E., Talbot's Bands and the

Resolving Power of Spectroscopes,

322.
Duane, William, The Measurement of "A"

by Means of X-Rays, 93.
The High Frequency Absorption Bands

of Some of the Elements, 98.
A Reactive Modification of Hydrogen

Produced by Alpha-Radiation, 116.
The Critical Absorption of Some of the

Chemical Elements for High Fre-
quency X-Rays, 697.
Djmamical-EIectrical Systems, Theory of

Variable, H. W, Nichols, 171.

Electrical Conductivity, The, of Sputtered

Films, Robert W. King, 291.
Electron Emission, The Loss of Energy of

Wehnelt Cathodes by, W. Wilson, 79.
Electron Theory, Unipolar Induction and

Electron, 591.
Electrons, The Passage of Low Speed,

through Mercury Vapor and the

Ionizing Potential of Mercury Vapor,

John T. Tate, 81.
Electrons, The Emission of, by a Metal when

Bombarded by Positive Ions in a

Vacuum, W. L. Cheney, 335.
Electrons, The Emission of, in the Selective

and Normal Photo-electric Effects,

A. LX, Hughes, 490.
Electrons. Total Ionization by Slow, J, B.

Johnson, 609.
Entropy, The Kinetic Theory of. W. P.

Roop, 83.
Errata, 787.
Expansion Theorem, On a General, for the

Transient Oscillations of a Connected
System, John R. Carson, 217.
Expansion, Thermal, of Marble. Uoyd if.
Schad, 74.

Ferromagnetic Substances. On a Molecular
Theory of, Kotaro Honda and JunMo
Okubo, 70.

Films. The Electrical Conductivity of Sput-
tered, Robert W. King, 291.

Fluorescence. The Wave-length of Light
from the Spark which Excites, in
Nitrogen, C. F. Meyer, 91.

Fluorescence, The, of Four Double Nitrates,
H. L. Howes and D, T, Wilber, 348.

Fluorescence, The Wave-Length of Light
from the Spark which excites, in Ni-
trogen, Charles F, Meyer, 575*

Fors3rthe, W. E., Color Temperature Scales
for Tungsten and Carbon, 395.

Gallium, The K Series of the X-Ray Spec-
trum of. H. 5. Uhler and C. D, Cooh*
sey, 645.

Gibbs, R. C, Note on the Absorption of
Tetrachlorophenolphthalein, 779.

Gibson, K. S., Ultraviolet and Visible Ab-
sorption Spectra of Phenolphthalein,
Phenolsulphonphthalein and Some
Halogen Derivatives, 767*

Gilbreath; J. A.. Ionization of Potassium
Vapor by Ordinary Light, 166.

Goucher, F. S.. Ionization and Excitation of
Radiation by Electron Impact in
Mercury Vapor and Hydrogen, loi.

Gravitation, Generalized Coordinates. Rela-
tivity and Gravitation. E. B. Wilson,

89.
Grondahl. L. O.. Experimental Evidence for

the Parson Magneton. 586.
Gusts. The Motion of an Aeroplane in, E. B,

Wilson, 89.

***,*• The Measurement of, by Means of X-
Rays, F. C. Blake and William Duane,

93.
'**." The value of. as Determined by Means

of X-Rays. F. C. Blake and WUliam

Duane, 624.
Hall Effect. The Reversal of the. in Alloys.

Alpheus W. Smith, 358-
Heaps, C. W.. Resistance and Magnetiza-
tion, 366.
Heat Convection in Air and Newton's Law

of Cooling, W. P. White, 743.
Helium, A Determination of the Density of,

by Means of a Quartz Micro-Balance,

r. S. Taylor, 653.
Helium, The Stark Effect in, and Neon,

Harry Nyquist, 226.
Hennings, A. E., The Energy of Emission of

Photo-Electrons from Film Coated

and Non-Homogeneous Surfaces: A

790

INDEX TO VOLUME X.

Theoretical Study, A, E, Hennings,

78.
The Significance of Certain New Phe-
nomena Recently Observed in Pre-
liminary Experiments on the Tem-
perature Co^cient of Contact Poten-
tial. 89.

Honda, Kotaro, On a Molecular Theory of
Ferromagnetic Substances, 705.

Howes, H. L., The Fluorescence of Four
Double Nitrates, 348.

Howe, H. E., Ultraviolet and Visible Absorp-
tion Spectra of Phenolphthalein,
Phenolsulphonphthalein and Some
Halogen Derivatives, 767.
Note on the Absorption of Tetrachloro-
phenolphthalein, 779.

Hoxton, L. G., A Measuring Engine for
Reading Wave-Lengths from Pris-
matic Spectrograms, 90.

Hughes, A. LI., The Emission of Electrons
in the Selective and Normal Photo-
electric Effects, 490.
The Ionizing Potentials of Gases. 495.

Hull, A. W.. A New Method of X-Ray Crys-
tal Analysis, 66 x.

Hyde, E. P., Color Temperature Scales for
Tungsten and Carbon, 395.

Hydrogen, Ionization and Excitation of
Radiation by Electron Impact in
Mercury Vapor and, Bergen Davis
and P. S. Goucher, loi.

Hydrogen, A Reactive Modification of, Pro-
duced by Alpha-Radiation, William
Duane and Gerald L. Wendt, 116.

Impact, Elastic, of Electrons with Helium

Atoms, J, M. Benade, 77.
Index, 788.
Induction, Unipolar, and Electron Theory,

George B. Pegram, 591.
Infra-red Absorption Spectra, On the Occur-
rence of Harmonics in the, of Gases,

W, W. CobUntM, 96.
Instability of Electrified Liquid Surfaces,

John Zeleny, x.
Interferometer, On the Limit of Interference

in the Fabry-Perot. Megh Nad Saha,

782.
Ions, The Emission of Electrons by a Metal

when Bombarded by Positive, in a

Vacuum. W. L. Cheney, 335.
Ionization. Theory of, by partially Elastic

Collisions. K, T. Compion, 80.
Ionization. Theoretical Considerations Con-
cerning, and "Single-Lined Spectra,"

H. J. Van der Bijl, 546.
Ionization, Total, by Slow Electrons, J, B.

Johnson, 609.
Ionization Potentials. On the. of Vapors and

Gases. J. C. McLennan, 84.
Ionization of Potassium Vapor by Ordinary

Light. J. A. Gilbreath, 166.
Ionizing Potentials. The, of Gases, A. U,

Hughes and A. A. Dixon, 495.

Ionization Potential, The, of Electrodes in
Various Gases. P. if. Bishop, 244.

Iron Oxide, Electrical and Thermal Proper-
ties of, C. C. BidwdL, 756.

Ishida, Yoshio, KineUc Theory of Rigid
Molecules, 305.

Johnson, J. B., Total Ionization by Slow
Electrons, 609.

Jones, Arthur Taber, Notes on Meld6's Ex-
periment, 541.

Jones, L. T., The Mercury-Arc Pump; The
Dependence of its Rate of Exhaustion
on Current, 301.

Kinetic Theory of Rigid Molecules, Yoshio
Ishida, 305.

King, Robert W.. The Electrical Conductiv-
ity of Sputtered Films, 29X.

Kunz, Jakob, Amplification of the Photo-
electric Current by the Audion, 205.

McKeehan, L. W., The Diffusion of Actin-
ium Emanation and the Range of
Recoil from it, 473-

McLennan, J. C, On the Ionization Poten-
tials of Vapors and Gases, 84.

Magie, William Francis, The Relation of
Osmotic Pressure to Temperature II,
64.

Magnetic Molecule, The Magnetization of
Iron, Nickel and Cobalt by Rotation
and the Nature of the, 5. J. BameU, 7.

Magneton, Experimental Evidence for the
Parson, L. 0. Grondahl, 586.

Magnetization, The, of Iron, Nickel and
Cobalt by Rotation and the Nature of
the Magnetic Molecule, 5. J, Barneti,

Magnetization, Resistance and, C. W,
Heaps, 366.

Magnetostrictive Effects, A Study of the
Joule and Wiedemann, in the Same
Specimens of Nickel, 5. R. WiUiams,
129.

Marble, Thermal Expansion of, Lioyd if.
Schad, 74.

Mathematical Structure, A New Theory
Concerning the, of Band Series, Ray-
mond 7. Birget 88.

Measuring Engine, A, for Reading Wave-
Lengths from Prismatic Spectro-
grams, L. G. Hoxton, 90.

Meld6's Experiment, Notes on, Arthur Taber
Jones and Marion Eveline Phelps, 541.

Mendenhall, C. E., A Determination of the
Planck Radiation Constant C2. 515.

Mercury Vapor, The Absorption of, by Tin-
Cadmium Alloy, L. A . Welo, 583.

Mercury- Arc Pump, The; The Dependence
of its Rate of Exhaustion on Current.
L. T. Jones and H. 0. Russell, 301.

VOL.X.1

No. 6. J

INDEX TO VOLUME X.

791

Meyer. Charles F.. The Wave-Lcngth of
Light from the Spark which Exdtes
Fluorescence in Nitrogen. 91.

Mercury Vapor, Ionization and Excitation
of Radiation by Electron Impact in,
and Hydrogen. Bergen Davis and F,
S. Goucher^ loi.

Meyer, Charles G, The Wave-Lcngth of
Light from the Spcu-k which Exictes
Fluorescence in Nitrogen, 575.

Millikan, R. A.. Radiation and Atomic
Structure, 194.

Molecular Theory, On a, of Ferromagnetic
Substances. Kotaro Honda and Junto
Okubo, 705.

New books. 214. 413. 589.

Neon. The Stark Effect in Helium and.
Harry Nyquist, 226.

Newton's Law. Heat Convection in Air and.
of Cooling, W. P. White, 743-

Nichols. H. W., Theory of Variable Dynam-
ical Electrical Systems. 171.

Nyquist. Harry, The Stark Effect in Helium
and Neon, 226.

Okubo, Junzo, On a Molecular Theory of
Ferromagnetic Substances, 705.

Oscillatory Spark Discharges between Un-
like Metals, D. L. Rich, 140.

Oscillating Systems Damped by Resistance
Proportional to the Square of the
Velocity. /. Parker Van Zandt, 415.

Oppitz, Louis K., Otpical Constants of the
Binary Alloys of Silver with Copper
and Platinum, 156.

Optical Constants by Reflection Measure-
ments, L. B. Tuckerman, Jr,, and
A. Q. Tool, 87.

Optical Constants of the Binary Alloys of
Silver with Copper and Platinum.
Louis K. Oppitt, 156.

Osmotic Pressure. The Relation of, to Tem-
perature. II. William Francis Magie,
64.

Parson Magneton. Experimental Evidence
for the. L. O. Crondahl, 586.

Pegram. George B.. Unipolar Induction and
Electron Theory. 591.

Phelps. F. P.. Natural and Magnetic Rota-
tion at High Temperatures. 90.

Phelps. Marion Eveline, Notes on Meld^'s
Experiment, 541.

Photo-Electrons. The Theory of Emission of.
from Film Coated and Non-Homo-
geneous Surfaces: A Theoretical
Study. A. E, Hennings, jS.

Photoelectric Cell, The Use of a Thomson
Galvanometer with a, W. W. Cob-
lenti, 97.

Photoelectric Current, Amplification of the,
by the Audion, Jakob Kunz, 205.

Photoelectric Effects. The Emission of
Electrons in the Selective and Normal.
A, LI. Hughes, 490.

Photometry of Lights, A Proposed Method
for the. of Different Colors. Irwin G,
Priest, 208.

Planck Radiation Constant C3. A Deter-
mination of the. C. E. MendenhaU,

515.
Planck's Radiation Law. The Necessary

Physical Assumptions Underlying a

Proof of. Russell v. Bichowsky, 92.
Polarization at the Cathode in Oxygen, C. A .

Skinner, 76.
Potential, Distribution of, in a Corona Tube.

Harry T. Booth, 266.
Potassium Vapor. Ionization of. by Ordinary

Light, J. A. Gilbreaih, 166.
Priest. Irwin G., A Proposed Method for the

Photometry of Lights of Different

Colors, III, 208.

Radiation, Ionization and Excitation of, by
Electron Impact in Mercury Vapor
and Hydrogen, Bergen Davis and F. S.
Goucher, loi.

Radiation, and Atomic Structure, R. A,
MiUikan, 194.

Reflection Measurements. Optical Constants
by, L. B. Tuckerman, Jr., and A. Q,
Tool, 87.

Reflection Coeflicient, The, of Monochro-
matic X-Rays from Rock Salt and
Caldte, A. H. Compton, 95.

Relativity. Generalized Coordinates, and
Gravitation. E. B. Wilson, 89.

Resistance and Magnetization, C. W. Heaps,
366.

Rich. D. L.. Oscillatory Spark Discharges
between Unlike Metals. 140.

Rotation. The Magnetization of Iron. Nickel
and Cobalt by. and the Nature of the
Magnetic Molecule. 5. J. Bameti, 7.

Rotation. Natural and Magnetic, at High
Temperatures, Frederick Bates and
F. P. Phelps, 90.

Roop. W. P.. The Kinetic Theory of En-
tropy. 83.

Russell. H. O.. The Mercury- Arc Pump; The
Dependence of its Rate of Exhaustion
on Current. 301.

Saha. Megh Nad. On the Limit of Inter-
ference in theFabry-Perot Interferom-
eter, 782.

Schad, Lloyd W., Thermal Expansion of
Marble, 74.

Shields. Margaret Calderwood, A Determina-
tion of the Ratio of the Specific Heats
of Hydrogen at i8®C, and I90**C.. 525.

Skinner. C. A.. Polarization at the Cathode
in Oxygen. 76.

Smith. Alpheus W.. The Reversal of the Hall
Effect in Alloys, 358.

792

INDEX TO VOLUME X.

Smith, Arthur Whitmore, Demagnetization
of Iron, 284.

Sound Intensity, A Condenser Transmitter
as a Uniformly Sensitive Instrument
for the Absolute Measurement of.
E. C. WenU, 22.

Specific Heats. A Determination of the Ratio
of the. of Hydrogen at i8®C. and
-ipo^C. Margaret Calderwood Shields,

525.

Specific Resistance, The. and Thermoelec-
tric Power of Metallic Calcium,
Charles Lee Swisher, 601.

Specific Volume, A Study of Apparent, in
Solution, Leroy D, Weld and John C.
Steinberg, 580.

Spectra, High Vacuum, from the Impact of
Cathode Rays, Louis Thompson, 207.

Speech, The Composition of, /. B. CrandaU,

74.

Spectra, Theoretical Considerations Con-
cerning Ionization and Single-Lined,
H, J. Van der Bijl, 546.

Spectra, Ultraviolet and Visible Absorption
of Phenolphthalein. Phenolsulphon-
phthalein and Some Halogen Deriva-
tives. 767.

Spectroscopes. Talbot's Bands and the Re-
solving Power of, Thomas E, Doubt,
322.

Stark Effect. The. in Helium and Neon,
Harry Nyquist, 226.

Steinberg, John C, A Study of Appcu-ent
Specific Volume in Solution, 580.

Strain. The Effect of, on Heterogeneous
Equilibrium. E. D. Williamson, 275.

Swisher, Charles Lee. The Specific Resis-
tance and Thermoelectric Power of
Metallic Calcium, 601.

Tate. John T., The Passage of Low Speed
Electrons through Mercury Vapor
and the Ionizing Potential of Mercury
Vapor. 81.

Taylor. T. S.. A Determination of the Den-
sity of Helium by Means of a Quartz
Micro-Balance, 653.

Temperature Coefficient, The Significance
of Certain New Phenomena Recently
Observed in Preliminary Experiments
on the. of Contact Potential, A, E.
Hennings, 89.

Temperature Scale. The True, of Tungsten
and its Emissive Powers at Incandes-
cent Temperatures, A, G. Worthing,

377.

Thermal Expansion, The, of Tungsten, at
Incandescent Temperatures, A, G.
Worthing, 624.

Thermoelectric Power, The Specific Resis-
tance and, of Metallic Calcium,
Charles Lee Swisher, 601.

Thermophone. The. as a Precision Source of
Sound. 22.

Thompson, Louis, High Vacuum Spectra

from the Impact of Cathode Rays,
207.
Thomson Galvanometer, The Use of a, with
a Photoelectric Cell. W, W. CoblenSs,

97.

Tin-Cadmium Alloy, The Absorption of
Mercury Vapor by, L. A, Welo, 583.

Tool, A. Q., Optical ConstanU by Reflection
Measurements. 87.

Tuckerman, L. B.. Jr., Optical Constants by
Reflection Measurements. 87.

Tungsten, The True Temperature Scale of,
and its Emissive Powers at Incandes-
cent Temperatures, A, G. Worthing,

377.

Tungsten, Color Temperature Scales for, and
Carbon, E. P. Hyde, F. E, Cody and
W, E. For sy the, 395-

Tungsten, The Thermal Expansion of, at In-
candescent Temperatures, A . G. Worth'
ing, 624.

Tyndall, E. P. T., Note on the Absorption of
Tetrachlorophenolphthalein. 779.

Uhler, R. S., The K Series of the X-Ray
Spectrum of Gallium, 645.

Vacuum Pump, The Parallel Jet High, TTO-
Ham W, Crawford, 557.

Van der Bijl, H. J., Theoretical Considera-
tions Concerning Ionization and
"Single-Lined Spectra." 546.

Van Zandt, J. Parker, Oscillating Systems
Damped by Resistance Proportional
to the Square of the Velocity, 415.

W. *

Warner, Earle H., The Pressure Increase in
the Corona, 483.

Wavelengths, A Measuring Engine for Read-
ing, from Prismatic Spectrograms,
L, G, Hoxton, 90.

Weld, Leroy D.. A Study of Apparent Specific
Volume in Solution, 580.

Weeks, Paul T., A Determination of the
Efficiency of Production of X-Rays,

564.
Wehnelt Cathodes. The Loss of Energy of,

by Electron Emission, W. Wilson, 79.
Welo, L. A., The Absorption of Mercury

Vapor by Tin-Cadmium Alloy, 583.
Wendt, Gerald L., A Reactive Modification

of Hydrogen Produced by Alpha-
Radiation, 116.
Wente, E. C, A Condenser Transmitter as a

Uniformly Sensitive Instrument for

the Absolute Measurement of Sound

Intensity, ij^. p^,,, V/
White, W. P., Heat Convection in. Air and

Newton's Law of Cooling, 743.
Wilber. D. T., The Fluorescence of Four

Double Nitrates. 348.
Williams. S. R., A Study of the Joule and

VOL.X.1

No. 6. j

INDEX TO VOLUME X,

793

Wiedemann Magnetostrictive Effects
in the Same Specimens of Nickel, 129.

Williamson. E. D., The Effect of Strain on
Heterogeneous Equilibrium, 275.

Wilson. W.. The Loss of Energy of Wehnelt
Cathodes by Electron Emission. 79.

Wilson, E. B.. The Motion of an A^oplane
in Gusts. 89.

Wilson. E. B., Generalized Coordinates,
Relativity and Gravitation, 89.

Worthing. A. G., The True Temperature
Scale of Tungsten and its Emissive
Powers at Incandescent Tempera-
tures. 377.
The Thermal Expansion of Tungsten at
Incandescent Temperatures. 624.

X-Ray Crystal Analsrsis. A New Method of,

i4. W, HuU. 661.
X-Ray Spectrum, The K Series of the. of

Gallium. H, S. VKler, and D. Cooksey

645.

X-Rays. The Measurement of ***** by means
of. F, C. Blake and William Duane, 93.

X-Rays. The Reflection Coefficient of Mono-
chromatic, from Rock Salt and Cal-
cite, A. H. ComptoHt 95.

X-Rays, An Experimental Investigation of
the Total Emission of. from Certain
Metals. C. S. Brainin, 461.

X-Rays. A Determination of the Efficiency
of Production of, Paul T. Weeks, 564.

X-Rays, The value of "A" as determined by
Means of, P, C. Blake and William
Duane, 624.

X-Rays. The Critical Absorption of Some of
the Chemical Elements for High Fre-
quency, P. C. Blake and William
Duane, 697.

Zeleny, John. Instability of Electrified
Liquid Surfaces, i.

•■r ^

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